Relativity for Physics Students
Uploaded by
krebilasRelativity for Physics Students
Uploaded by
krebilasLLJ<
CQ
OU
166116
73
American State Government
AMERICAN STATE
GOVERNMENT
By
W.
BROOKE GRAVES
CHIEF OF THE STATE LAW SECTION
LEGISLATIVE REFERENCE SERVICE
LIBRARY OF CONGRESS
wrTHIRD EDITION
D. G.
HEATH AND COMPANY
Boston
THE THEORY OF
RELATIVITY
BY
C.
M0LLER
PROFESSOR OF MATHEMATICAL PHYSICS
IN THE UNIVERSITY OF COPENHAGEN
OXFORD
AT THE CLARENDON PRESS
Oxford University Press, Amen House, London E.C. 4
GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON
BOMBAY CALCUTTA MADRAS KARACHI CAPE TOWN 1DADAN
Geoffrey Cumberlege, Publisher to the University
FIRST EDITION 1952
REPRINTED LITHOGRAPHICALLY IN GREAT BRITAIN
AT THE UNIVERSITY PRESS, OXFORD
FROM CORRECTED SHEETS OF THE FIRST EDITION
1955
PREFACE
THE
present
monograph
is
a somewhat extended version of a course of
which I have given at the University of Copenhagen during the
twenty years. Consequently, it is primarily a textbook for students
in physics whose mathematical and physical training does not go
beyond the methods of non-relativistic mechanics and electrodynamics.
The intention has been to give an account of what may be called the
classical theory of relativity in which all quantum effects are disregarded.
In view of the paramofintx importance of quantum phenomena in
lectures
last
modern
*^
?*
f^
physics, the limitation of the subject to classical phenomena
be
considered a serious defect of the book. However, there are
might
several important reasons for such a limitation of the subject. At
present, a complete self-consistent reiativistic
quantum theory does
Moreover, the classical theory of relativity, which by itself
an
gives
admirably precise description of a very extended field of
physical phenomena, must be the starting-point for the future developnot exist.
ment of a consistent reiativistic quantum theory. For a student and
research worker in this field an intimate aquaintance with the principles and methods of the classical theory of relativity is, therefore,
just as indispensable as is the knowledge of the methods of Newtonian
mechanics for a real understanding of ordinary quantum mechanics.
this, the classical theory of relativity is one of the most
and
beautiful parts of theoretical physics on account of its
fascinating
inner consistency and the simplicity and generality of its basic assump-
Apart from
tions.
The presentation of the subject in the present volume differs somewhat from the usual one in that the four-dimensional formulation of
the theory plays a less dominant role than in most of the current textCertainly the four-dimensional representation, which is based
on the symmetry between the space and time variables revealed by the
books.
discovery of the Lorentz transformation, is the most elegant way of
expressing the principle of relativity in mathematical language^ and it
has been of the utmost importance for the rapid development, particularly of the general theory of relativity. In the early books on relativity
it
was, therefore, quite natural to emphasize as strongly as possible
newly discovered similarity between the space and time variables.
this
However, in a textbook of today I think it is useful to stress again the
fundamental physical difference between space and time, which was
PREFACE
vi
somewhat concealed by the purely formal four-dimensional representation.
three chapters we have, therefore, avoided any reference to the four-dimensional picture, and the kinematics and point
mechanics of the special theory of relativity are fully developed by means
In the
first
of the usual three-dimensional vector calculus.
But
in the following
chapters also, where the elegant methods of the four-dimensional tensor
calculus are developed and applied, a three-dimensional formulation,
which gives a better insight into the physical meaning of the theory, is
frequently given. As an example, T shall mention the treatment in
and 111 of a
The motion of the
1
10
freely falling particle in a given gravitational field.
particle is, of course, completely described by the
statement that the time track of the particle in 4-space is a geodesic
line, but this quasi-geometrical description does not convey a physical
understanding of the phenomenon. In the three-dimensional physical
space, however, the motion of the particle can be described by an
equation of motion of the same type as that for a particle subject to an
arbitrary force in a system of inertia, the only difference being that the
geometry in the physical space is in general non-Euclidean. In this
way we
obtain definite expressions for the gravitational force on the
particle as well as for the mass, momentum, and total energy of a
particle moving with arbitrary velocity in a given gravitational field.
The three-dimensional point of view thus
leads to a reintroduction of
dynamical concepts into the gravitational theory, which,
makes
it
I believe,
easier for the student fully to grasp the physical content of
the general theory of relativity.
Since a real understanding of a physical theory is possible only
through an intimate knowledge of its predecessors, the whole of
has been devoted to an historical survey of the difficulties of
Many students who intend to specialize
in experimental physics may feel that the time and effort which are
Chapter
the non-relativistic theories.
needed to learn the methods of the general tensor calculus are out of
proportion to the use they can make of this formalism in their future
work. Such readers will find the main results of the special theory of
relativity in the first thirty -eight sections. They will also be able to
read Chapter VIII and in this way obtain an insight into the ideas
underlying Einstein's general theory of relativity without spending any
time on the laborious task of learning its special mathematical methods.
We have included only those developments of the theory of relativity
which can be regarded as safely established, the various attempts at
PREFACE
vii
constructing a unified theory of gravitation and electromagnetism
falling outside the scope of the present book. Also the cosmological
problems have only been touched upon, since these problems have been
extensively treated
these restrictions it
by Tolman
in this series of
monographs. Within
hoped, however, that the reader will find a fairly
complete and well-rounded account of one of the most beautiful chapters in the history of science, which for the main part was written by
is
a single man, Albert Einstein.
On completion of this work
gratefully acknowledge the help and
many quarters. First of all I want
advice which I have received from
my
to express
interest in my
deep gratitude to Professor Niels Bohr for his kind
work during
all these years and for the constant inspirafrom
tion derived
many discussions and conversations at his institute.
My thanks are due to Professor N. F. Mott and Professor I. N. Srieddon
for reading the manuscript and eliminating the worst danicisms. I also
wish to thank the staff of the Clarendon Press for their friendly co-
operation.
am
W. Kohri and Dr. W. J. Swiatecki for many
which
have
considerably improved the text and, in particular,
suggestions
to mag. scient. J. Lindhard who has been of great help in checking
all the equations and reading the proofs.
Finally, I am grateful to
I
indebted to Dr.
Hellmann for her untiring assistance
manuscript and the proof-reading.
Miss
S.
in the preparation of the
C.
COPENHAGEN
November 1951
M.
CONTENTS
CHAPTER
I.
....
THE FOUNDATIONS OF THE SPECIAL THEORY OF
RELATIVITY. HISTORICAL SURVEY
1.
2.
3.
4.
The
The
relativity principle of mechanics.
special principle of relativity
The Galilean transformation
9.
Lorentz's theory of electrons
7.
10.
10
_
.15
....
17
20
all effects
The aberration
of light
principle of relativity
order. Format's principle
... ...
...
of the
first
Michelson's experiment
13.
The contraction hypothesis
%8
.
Validity of the principle of relativity for
CHAPTER
II.
all
RELATIVTSTIC KINEMATICS
Simultaneity of events
16.
18.
The relativity of simultaneity
The special Lorentz transformation
The most general Lorentz transformation
19.
Contraction of bodies in motion
20.
The retardation of moving
clocks.
The
Transformation of particle velocities
28
29
41
44
clock paradox
'
48
51
The Thomas precession
Transformation of the characteristics of a wave according to the
theory of relativity
25
26
36
Successive Lorentz transformations.
25.
22
.31
.31
.33
21.
24.
phenomena
22.
23.
physical
15.
17.
Agreement between the ether theory and the
12.
14.
.4
.6
as regards
ll.
...
8.
6.
In variance of the phase of a plane wave
Transformation of the characteristics of a plane wave.
The Doppler effect
The velocity of light in vacuo
The velocity of light in refractive media
Hoek's and Fizeau's experiments
5.
53
.56
.58
The ray velocity in moving bodies
The Doppler effect, the aberration of light, and the dragging phenomenon according to the theory of relativity
.
.62
.67
.67
26.
III. RELATIVISTIC MECHANICS
Momentum and mass of a particle
27.
Force, work, kinetic energy
28.
29.
Transformation equations for momentum energy and force
Hyperbolic motion. Motion of an electrically charged particle in a
constant magnetic field
30.
Equivalence of energy and mass
CHAPTER
....
.
70
.
71
.74
.77
CONTENTS
31.
Inelastic collisions.
ix
Mass of a closed system of
particles
Experimental yenficatioii of relativistic mechanics
CHAPTER IV. FOUR -DIMENSIONAL FORMULATION
32.
34.
82
85
OF THE
THEORY OF RELATIVITY: TENSOR CALCULUS
33.
dimensional representation
36.
37.
Four-velocity and
velocity
acceleration.
Wave-number
...
vector.
92
92
.96
Covanance of the laws of nature in four-dimensional formulation
The four-dimensional line element or interval. Four-vectors
35.
Four -dimensional representation of the Lorontz transformation
Lorentz contraction and retardation of moving clocks in four-
97
99
Four-ray
101
^38. Four-momentum. Four -force. Fundamental equations of point
/
x
39.
's 40.
mechanics
in four -dimensional vector
Tensors of rank 2
Angular
form
momentum and moment
representation
'41, Tensors
42.
Pseudo-tensors
.104
.108
of force in four-dimensional
.
...
of arbitrary rank
43.
.110
.111
.112
.113
.114
44.
The Levi-Civita symbol
Dual tensors
45.
Infinitesimal Lorentz transformations.
46.
Successive Lorentz transformations
47.
Successive rest systems of a particle in arbitrary rectilinear motion
and in constant circular motion
.121
without rotation
Lorentz transformations
Tensor and pseudo -tensor fields. Tensor analysis
49. Gauss's theorem in four-dimensional space
48.
50.
51.
The fundamental equations of mechanics
The kinetic energy-momentum tensor
RAFTER V.
for incoherent
.
ELECTRODYNAMICS IN THE VACUUM
.117
.118
matter
.
.125
.128
.
52.
The fundamental equations
53.
Covanance of the fundamental equations of electrodynamics
under Lorentz transformations. The electromagnetic field
54.
The
tensor
.139
........
four-potential.
Gauge transformation
Four-dimensional integral representation of the four-potential
56. Retarded potentials. Lienard-Wiechert's potentials for point charges
55.
58.
The field of a uniformly moving point charge
The electromagnetic forces acting on charged matter
59.
Variational principle of electrodynamics
60.
The electromagnetic energy -momentum tensor
The total energy-momentum tensor
57.
61.
139
vacuum.
of electrodynamics in the
Four-current density for electric charge
130
.136
141
143
144
147
.151
.154
.157
159
.161
CONTENTS
GENERAL CLOSED SYSTEMS. MECHANICS OF
THEORY
.163
CHAPTER VI.
.....
ELASTIC CONTINUA. FIELD
62.
Definition of a closed system
63.
Four -momentum
system
arid angular
.
momentum
Centre of mass
65.
The fundamental equations of mechanics
66.
Transformation of
67.
Perfect fluids
68.
Scalar
density
.166
170
contmua
momentum density, and energy
elastic stress,
.
in elastic
.179
fields.
General
field
theory
...
General properties of non-closed systems
70. Static non-closed systems
69.
Electrostatic systems.
72.
The fundamental equations
matter.
73.
Mmkowski's
74.
The
field
Classical
....
of
equations
electrodynamics
.
m uniformly moving
in
stationary
bodies
constitutive equations in four-dimensional language.
.
.195
.196
Boundary
.201
Electromagnetic energy -moment urn tensor arid four-force density
76. The propagation velocity of the energy of a light wave in a moving
.....
75.
refractive body
The laws of thermodynamics
matter
Transformation properties of the thermodynamical quantities
Four/dimensional formulation of the laws of thermodynamics
80.
Ideal
81.
Black-body radiation
monatomic gases
....
.
206
212
214
.215
216
THE FOUNDATIONS OF THE GENERAL THEORY
.218
OF RELATIVITY
*82. The
202
.211
79.
CHAPTER VIII.
in stationary
188
.188
.191
.192
models of the electron
conditions
78.
181
.184
71.
77.
173
NON-CLOSED SYSTEMS. ELECTRODYNAMICS IN
AND PARAMAGNETIC SUBSTANCES. THERMO-
DIELECTRIC,
DYNAMICS
meson
CHAPTER VII.
163
four-tensor for a closed
64.
.....
.....
general principle of relativity
83.
The
84.
Uniformly rotating systems of coordinates. Space and time
principle of equivalence
general theory of relativity
218
220
in the
222
85.
Non-Euclidean geometry. The metric tensor
86.
Geodesic lines
87.
Determination of the metric tensor by direct measurements.
.231
Geometry in n-dimensional space
88.
General accelerated systems of reference.
missible space -time transformations
....
226
.228
The most general ad233
CONTENTS
xi
89.
Space and time measurements in an arbitrary system of reference.
Experimental determination of the functions g lk
90.
The spatial geometry in the rotating system of
The time tracks of free particles and light rays
91.
92.
93.
....
The dynamical gravitational potentials
The rate of a moving standard clock in a
gravitational field
237
240
244
245
247
95.
Transformation of coordinates inside a fixed system of reference
Further simple examples of accelerated systems of reference
96.
Rigid systems of reference with an arbitrary motion of the origin
253
97.
Rigid frames of reference moving in the direction of the X-axis
255
98.
The clock paradox
94.
reference
248
250
.258
PERMANENT GRAVITATIONAL FIELDS. TENSOR
CALCULUS IN A GENERAL RIEMANNIAN SPACE
CHAPTER IX.
Four-dimensional formulation of the general principle of relativity
and of the principle of equivalence
100. Contra variant and covanant components of a four-vector
264
99.
Tensor algebra
102. Pseudo -tensors. Dual tensors
101.
formulae
103.
Geodesic
104.
Local systems of inertia
105.
Parallel displacement of vectors
lines.
Christoffel's
.
The contracted forms of the curvature tensor
266
270
272
.274
.276
.279
......
.
264
.269
Tensor analysis. Co variant differentiation
107. The curvature tensor
106.
108.
284
286
THE INFLUENCE OF GRAVITATIONAL FIELDS ON
PHYSICAL PHENOMENA
.288
CHAPTER X.
109.
Mechanics of
110.
Momentum and mass
111.
Total energy of a particle in a stationary gravitational
General point mechanics
112.
1
13.
free particles in the presence of gravitational fields
of a particle. Gravitational force
field
288
290
294
.....
.
295
Time-orthogonal systems of coordinates. Elimination of the dynamical potentials
.296
.
114.
Mechanics of continuous systems
115.
The electromagnetic
field
equations
principle
.......
Electromagnetic force and energy-momentum tensor
117. Propagation of light in a static gravitational field.
116.
298
302
305
Format's
308
THE FUNDAMENTAL LAWS OF GRAVITATION IN
THE GENERAL THEORY OF RELATIVITY
.310
CHAPTER XI.
....
.
118.
119.
The
The
gravitational field
linear
equations
approximation for weak
fields
310
.313
CONTENTS
xii
120.
Simple applications
of the linear
equations for weak
and Coriohs
relativity of centrifugal forces
fields.
forces
The
.315
121. Equivalent systems of coordinates. Systems with spherical sym-
metry
Static systems with spherical
123.
Schwarzsch ild's exterior solution
.321
....
122.
symmetry
124.
Schwarzschild's solution for the interior of a perfect fluid
125.
The variational principle for gravitational fields
The laws of conservation of energy and momentum
Different expressions for the densities of energy and momentum
The gravitational mass and total energy and momentum of an
126.
127.
128.
328
333
337
......
isolated system
323
325
EXPERIMENTAL VERIFICATION OF THE GENERAL
THEORY OF RELATIVITY. COSMOLOGICAL PROBLEMS
341
342
CHAPTER XII.
131.
The gravitational shift of spectral lines
The advance of the perihelion of Mercury
The gravitational deflexion of light
132.
Cosrnological models
133.
The Einstein universe
The de Sitter universe
129.
130.
134.
348
353
APPENDIXES
357
....
362
371
1.
Gauss's theorem
The transformation equations for the four-cm rent density
Plane waves in a homogeneous isotropie substance
3.
5.
6.
7.
8.
371
lic ,
....
.
372
373
Transformation of the gravitational field variables y y ^,
change of coordinates inside a definite system of reference
Dual tensors in a three -dimensional space
a> lK
t,
by a
.
374
375
376
The condition for flat space
The action principle and the Hamiltoman equations for a particle
378
in an arbitrary gravitational field
The connexion between the determinants of the space-time metric
.381
tensor and the spatial metric tensor
m
The derivatives of the function with respect to g%* and c? and some
.....
.
9.
356
2.
4.
346
346
SUBJECT INDEX
......
....
identities containing these derivatives
AUTHOR INDEX
.382
384
385
THE FOUNDATIONS OF THE SPECIAL THEORY
OF RELATIVITY. HISTORICAL SURVEY
The
1.
relativity principle of
mechanics. The Galilean trans-
formation
THE special theory of relativity which was developed in the beginning of
the twentieth century, especially through Einstein's work, has its roots
far back in the past. In a way, this theory can be regarded as a continuaand completion of the ideas which have been the basis of our descripand Newton. The basic postulate
of this theory, the so-called special principle of relativity,! had already
in Galileo's and Huyghens's works played a decisive role in the development of the fundamental laws of mechanics. Also the validity of the
tion
tion of nature since the times of Galileo
principle of relativity for the phenomena of mechanics is a simple
consequence of the Newtonian laws of mechanics. Since the laws of
mechanics are especially well suited for the illustration of the principle
we shall start by considering purely mechanical phenomena.
of relativity,
According to Newton's
first law, the law of inertia, a material particle
to itself will continue to move in a straight line with constant
velocity. Since one cannot simply speak of motion, but only of motion
relative to something else, this statement has a precise meaning only
when
left
when a
certain well-defined system of reference has been established
which the velocity of the particle is assumed to be measured.
relative to
Therefore
Newton introduced the notion
of the 'absolute space', repre-
senting that system of reference relative to which every motion should
be measured. Experience shows that the fixed stars as a whole may
be regarded as approximately at rest relative to the 'absolute space',
body sufficiently far away from celestial matter always moves with
uniform velocity relative to the fixed stars.
for a
however, obvious that the law of inertia holds also in every other
rigid system of reference moving with uniform velocity relative to the
'absolute' system, for a free particle will also be in uniform translatory
It
is,
motion with respect to such a system. All systems of reference for which
the law of inertia is valid are called systems of inertia. They form a
When
reference is made to the principle of relativity
Chapters I-VII we always
view the principle of special relativity as contrasted with the principle of general
relativity which is the basis of the general theory of relativity.
t
have
in
3595.60
r>
FOUNDATIONS OK SPECIAL THEORY OF RELATIVITY
I,
threefold infinity of rigid systems of reference mo\ing in straight lines
relative to each other. One of them is the
and with constant velocity
absolute system which is at rest relative to the fixed stars as a whole;
but as regards the validity of the law of inertia, all systems of inertia are
completely equivalent.
principle of relativity in mechanics states that the systems
of inertia are also completely equivalent with regard to the other laws of
Now, the
mechanics. If this
is
true
all
mechanical phenomena will take the same
course of development in any system of inertia so that it is impossible
from observations of such phenomena to detect a uniform motion of the
system as a whole relative to the 'absolute' system. Thus, a study of
mechanical phenomena alone can never lead to a determination of the
'absolute' system.
We
now
fundamental equations of Newtonian
mechanics actually are in accordance with the principle of relativity.
Let us consider two arbitrary systems of inertia, / and 1' In each of
shall
see that the
'
use definite systems of coordinates 8 and #'.
may, for instance, choose Cartesian coordinates x
(x,y,z) and
the
to
/
and
in
1',
According
conceptions
respectively.
(jc',y',z'}
these frames of reference
We
x'
we
of space and time, derived from our usual experience, which also form
the basis of the Newtonian formulation of the fundamental laws of
mechanics, the connexion between the coordinate vectors x and x' for
one and the same space point in the two coordinate systems JS and $'
by
where v
is
relative to
'
;
a vector denoting velocity and direction of motion of ft'
/ is the time and, for the sake of
simplicity, it is assumed
*V.
that the origins of the two systems of coordinates coincide at the time
-- 0. To the
equations (1) may be added the equation
t'
(16)
which states that the parameter describing the time is the same in all
systems of inertia. Thus, in the Newtonian description of physical
phenomena the time is an absolute quantity. The equations (1(7) and
(l/>)
are often referred to as the Galilean transformation.
two systems of coordinates are
If the directions of the axes of the
v has the direction of the x-axis, we obtain a special
Galilean transformation which can be written
parallel,
and
if
I,
HISTORICAL SURVEY
Since the systems of coordinates 8 and 8' are completely equivalent, at
any rate as far as kinematics is concerned, and*since S obviously moves
and
v relative to S
with the velocity
the inverse transformations to
are simply obtained by interchanging the primed
v.
unprimed variables and simultaneously replacing v by
(2)
now
consider an arbitrary motion of a material particle.
differentiation of (1 a) we get
Let us
dx' _
~~
Jx
dt~
"eft'
or
u'
where u and
(1)
and the
By
v,
(3)
u' represent the velocities of the particle in the
two systems
the usual addition theorem of velocities. For a special
Galilean transformation (2), (3) reduces to
of inertia.
(3) is
ux
When
ux
uy
v,
uy
uz
the velocity vector u, arid thus also u',
may be written
is
uz
(4)
perpendicular to the
2-axis, (4)
u'
u'
sm&'
u$in&,
where $ and &' are the angles between the a;-axis and the directions of u
and u', respectively. Further, u ~ |u u' -- |u' denote the magnitudes
of the vectors u and u'. If we now divide one of these equations by the
|,
other we obtain
tan,?'
= --
COS IT"--V,
V/U
(5)
and by summation of the squares of the equations we get
u'
r
l
]_2-
co8* +
?;
^
U")
-Y
(6)
Now let us assume that the material particle with the mass m is acted
on by a force F. In the absolute system of coordinates S the particle
will then obtain an acceleration given according to Newton's second law
by the equation
From
(1 a)
and
(16)
it
now
follows that
&* dt'*
^
dt
'
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
and
Newtonian mechanics
in
.since
quantities,
i.e.
we obtain
w'
p
r/
x'
------
a/
Thus we
and masses are absolute
forces
m/
=^
F'.
see that the second law of
I,
(g)
(10)
Newton
is
valid in every system
of inertia in accoi dance with the principle of relativity. This can be
expressed more accurately by stating that the Newtonian fundamental
equations are invariant under (Jalilean transformations. As is well
known, this in variance does not hold for nioie general transformations
leading to accelerated systems ot reference. If one wants to treat
mechanical phenomena in such systems, one has to introduce extra
and
which only
depend on the acceleration of the frame of reference and therefore are in
no causal relationship with the physical properties of other terrestrial
systems. It was just this difference between the uniformly moving and
the accelerated systems of reference which led Newton to the conception
fictitious forces, e g. centrifugal forces
Coriolis forces
of absolute space.
2.
The
special principle of relativity
As already mentioned, the
validity of the principle of relativity in
mechanics prevents a unique determination
reference from studies of mechanical
ot the absolute
phenomena
assumption of the special theory of relativity
is
alone.
Now
system of
the basic
that the special principle
According to this theory,
physical phenomena should have the same course of development in
of relativity
is
valid for all physical laws."\
all
all
systems of inertia, and observers installed in different systems of inertia
should thus as a result of their experiments arrive at the establishment of
the same laws of nature.
the notion of absolute space obviously loses its meaning,
any system of inertia with equally good reason can claim to be the
absolute system of reference. Of course nobody can prevent us from
It this is so,
since
calling one definite system of inertia, e g. the one which is at rest relative
to the fixed stars, the absolute system and expressing all laws of nature
Such a procedure is, however, extremely
view of the arbitrariness in the choice of the absolute
in coordinates of this system.
unsatisfactory in
system. It
is,
furthermore, very inconvenient to proceed in this manner.
The physical experiments from which the laws of nature are derived
usually not performed in a system of reference which
is
are
at rest relative
f With tho exception of the la\vn of gravitation which find then natural place in the
general theory of lelativity.
HISTORICAL SURVEY
I,
to the fixed stars.
On account of its motion around the sun the earth will
in the course of a year represent widely different systems of inertia if we
disregard the small acceleration of the earth in this motion. The trans-
formation to the coordinates of the 'absolute' system
is
therefore rather
complicated.
The validity of the principle of relativity for all physical phenomena
now makes such a transformation unnecessary, since the system of
is at rest at the moment considered is equiyaof
inertia. This obviously leads to an enormous
other
system
any
our
in
description of nature.
simplification
inertia in
which the earth
lent to
However, this simplification has to be paid for, as we shall now see, by
an abandonment of our usual notions of time and space. The extension
of the principle of relativity to electromagnetic phenomena means, as
mentioned before, that physicists who have established their laboratories
two different systems of inertia will, as a result of their experiments,
be led independently to Maxwell's fundamental equations of electrodynamics. These equations contain a universal constant c which can
in
be determined by means of purely electromagnetic measurements, and
10
cm. /sec. f On the other hand, it is a
is very closely equal to 3xl0
Maxwell's
of
equations that electromagnetic waves
simple consequence
in empty space propagate with the velocity c, independently of the way
in
which they are created. Since
light waves, according to Maxwell's
theory of light, are special electromagnetic waves, the velocity with which
light is propagated in vacua must also be independent of the state of
motion of the
light source
and equal to the constant c.
If Maxwell's equa-
tions in accordance with the relativity principle are valid in
any system
of inertia, the velocity of light must have the same constant value c in
all systems of inertia, independently of the motion of the
light source.
This
is
obviously in conflict with the usual kinematical concepts accord-
ing to which we should expect, for instance, to find a lower velocity of
f} if the relative motion of >V with
light in X' than
respect to H has the
same
direction as the direction of propagation of the light ray.
Consequently the acceptance of the relativity principle must necessarily lead to a revision of our ordinary concepts of space and time.
Before taking such a radical step one would naturally w ant to be sure
r
that
it is
really necessary. This question
can only be settled as a result of
experiment. Optical experiments are especially suited to this purpose
in view of the high accuracy obtainable with optical instruments. In
t
W. Wober und
schaften,
No. 142.
R. Kohlrausch (1856), Ostwalds Klassiker der exakten
Wis&en-
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
I,
the following section we shall therefore give a short historical survey of
the numerous optical experiments which have been performed in an
attempt to detect effects depending on the motion of the apparatus with
respect to an 'absolute' space. These experiments all gave negative
results and finally led to a general acceptance of the principle of relativity
.
3.
Invariance of the phase of a plane wave
In contradistinction to the relativistic standpoint according to which
Maxwell's equations are valid in any inertial system, Maxwell and his
contemporaries maintained that the fundamental equations of electrodynamics were valid only in one system of inertia, \ iz the system which
is
The ether was imagined
matter and empty space and
at rest relative to the so-called 'world ether'.
as a
medium which
all
penetrates through
and electromagnetic phenomena
Moreover, the ether was supposed to lepresent the absolute system of
reference, thus giving a substantial physical meaning to Newton's notion
which was the
carrier of all optical
of absolute space. In the present section we shall fully adopt this point
of view, and our first task will be to see what consequences this will have
for the course of development of optical
moving
Let
phenomena in a system
of inertia
relative to the ether.
8 be
Relative to
a Cartesian coordinate system which
Ft,
is
at rest in the ether.
a plane monochromatic light wave in empty space will
10
3 X
cm /sec. A wave of this type
have the propagation velocity c
is completely determined by the phase velocity, the frequency of the
wave, and the direction of propagation. In the first place, we shall find
1
the transformation of these three quantities by a transition to a coordinate system *S' moving relative to the ether with a constant velocity
v in the direction of the .r-axis.
For simplicity
the
.r//-plane.
let
Then
us
assume that the normal of the vvavo plane lies
wave is described in N by a wave function
in
the*
ifj
.4
cos27r/<
(11)
where v is the frequency and ex is the angle between the wave normal n
and the .s-axis. /
.rcosa+j/sincx is the distance from the origin
to that wave plane which contains a point p with the coordinates (r, y)
(cf.
Fig/1).
The phase
in (11)
has the following simple physical meaning.
at the time
Let us assume that the wave crest which passes the origin
HISTORICAL SURVEY
I,
is provided with a label. Now
suppose an observer to be placed at
the point p who at the moment when the labelled wave arrives at p
begins to count the waves passing over the point p. The number of waves
t
counted by the observer
up to the time t will then just be equal to the phase F.
waves arrive per second and, since the labelled wave takes l/c
seconds to move from O to p, the observer is counting during an interval
In
of
fact, v
(1/c)
seconds.
Fir;. 1.
Now let 8' be the moving system introduced above, and let us assume
that the two coordinate systems 8' and 8 coincide at the time t when the labelled \vave passes the common origin of the systems. If;/
a point in 8' with coordinates (x\ ?/), which coincides
with;? at the time
t', the number of waves passing // trom the time of arrival
at;/ of the
wave
to
the
t
lime
will
of
course
be
the
same number F as
labeljed
up
is
t
On
the other hand, this number counted by an observer at
p'
will by a similar argument to that
applied in 8 be equal to
before.
F^
v'\t'
~V
,i
x cos a
)-?/
sin a
c'
where the primed letters in (12) denote the same physical quantities as
the corresponding unprimed letters in (11), but now measured in the
system of coordinates S' Thtfb the phase F is an invariant.
.
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
4.
I,
Transformation of the characteristics of a plane wave
The connexion between the coordinates (.r, ?/, /) in (11) and the
ordinates
tion
(2),
the two
(x,y,t)
(#', ?/,
in (12) is obviously given
co-
by the Galilean transforma-
since the pointy/ coincides with p at the time t'
t.
Equating
and
for
and
the
coordinates
(12)
expressions (11)
eliminating
by means of
(1)
we obtain
a;'
Now
cos
'+?/' sin
nA
equation must hold for all values of the independent variables
and this is possible only when the coefficients of these variables
are equal on both sides of the equation (13). Consequently we get the
this
x', y',
t',
equations
vll
c
v Sin
From
(14)
COS a
v COS
(is)
c\
equations (15) we get at once
tan
QL
a'
i.e.
"
solving the last equation for
of the wave, viz.
(16)
a,
(17)
(18)
c!
we
find
by means of
c~ vcosoc.
c'
The equations
tan a,
~ - ~.
and, further,
By
cosaj,
(14) that
(19)
show how the three characteristics
the frequency, the direction of the wave normal, and
(14), (17),
and
(19)
the phase velocity, will change at the transition to a coordinate system
in uniform motion with respect to the ether. Equation (17) shows that
the direction of the wave normal
is
the same in both systems of inertia.
On
the other hand, if a -/ \TT, the equations (14) and (19) involve the
velocity v, so that a measurement of frequency and velocity in principle
should be suited to determine the motion of the laboratory system with
respect to the ether. In the following we shall discuss these two effects
separately.
5.
The Doppler
effect
Equation (14), which
Doppler effect for
called
the mathematical expression of the solight waves, gives the connexion between the
is
HISTORICAL SURVEY
I,5
frequency v' in a moving frame of reference and the 'absolute' frequency
observed by an observer at rest in the ether. If n denotes a unit
v as
vector in the direction of the
wave normal, and v
is
the velocity vector
of S' relative to the ether, (14) can also be written
(20)
where n. v
appears
is
the scalar product of the two vectors. The Doppler effect
the observer is moving relative to the source of light.
when
However, the formula (20) cannot be used directly in such a case, since
as a rule both the observer and the source of light will have a motion
relative to the ether.
If v
is
the proper frequency of the light source,
the frequency measured by an observer at rest relative to the source
i.e.
of light,
we have
in analogy to (20)
v-Kl--n.v/c),
where v
By
(21)
the velocity of the source relative to the ether.
elimination of the unknown absolute frequency v we obtain from
is
l-(n.v)/c
The frequencies v and i>, the direction of propagation of the light n,
and the relative velocity v r
v v of the observer with respect to the
source of light can be determined directly by experiments, and (22) then
permits in principle a determination of the absolute velocities v and v
of the light source and the observer.
Both v and v are, however, very small compared with r, so that we
may
If
we
perform an expansion of (22) in terms of the small quantity (vn)/c.
neglect all terms oi higher than second order in v/c and v/r, we get,
introducing the relative velocity v r
v,
(n^)_(n^^
c2
In
iirst
approximation, the Doppler effect thus depends only on the
v of the light source enters
relative velocity v r The absolute velocity
only in the small second-order terms.
.
The Uoppler effect is observed in the spectra of the stars, the lines of
the spectrum being shifted towards violet or red according as the earth,
during its annual motion, moves nearer to or away from the observed star.
The
velocity of the earth in its orbit
is
approximately
3x
10 6 cm./sec.
10
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
and cosmic
I,
mostly of the same order of magnitude. Con10~ 4 i.e. the terms of the second order will be
sequently,
v/c
of the order of magnitude 10~ 8 which is far beyond the precision of such
velocities are
we have
measurements.
The Doppler
been observed also in the light from moving
measuring the frequency of the light emitted by
effect has
terrestrial sources.
By
rapidly moving hydrogen molecule ions in a positive ray tube, J. Stark f
found good agreement with the formula (23) as regards the terms of the
order. In these experiments the relative velocity v r and, thus, also
were of the order of 10 8 cm. /sec., i e. r/c
Also in this case, the
gJ
first
&
second-order terms were too small to be measured, and these experiments did not therefore allow a determination of the absolute velocity.
Much later, in the thirties, such experiments were repeated by I vest
with an improved experimental technique, which allowed the secondorder terms to be determined. The results obtained were not in agreement
with (23), but they agreed with a formula derived from the theory of
relativity (equation (II. 90), ('hap. II, 25). The second-order term was
found to be independent of the direction of the light emitted and dependent only on the relative velocity v r No motion relative to the ether could
.
be observed, in agreement with the principle of relativity.
These experiments, which were performed much later, had of course
no influence on the historical development of the relativity theory.
Because of the limited accuracy, the experiments by Stark did not allow
any decision to be formed on the validity of the principle of relativity,
but on the other hand the results of these experiments were not in contradiction with the principle of relativity.
The
velocity of light in vacua
We now turn to the question whether a measurement of the velocity
of light by terrestrial methods can lead to a determination of the absolute
6.
velocity v of the earth. Since v enters into equation (19), which can also
be written
c'
/0/1X
(n.v),
this should be possible in principle, as
mentioned on
(24)
p. 8.
The
well-
known measurements
of the velocity of light by Fizeau (1849) and Foucault (1865) showed, however, no influence at all of the motion of the
The velocity of light was always found to be the same, in agreement
earth.
f J. Stark, Ann. d Phys 21, 40 (1906), J. Stark and K Siogel, ibid 21, 457 (1906),
J Stark, W. Hermann, and S. Kmoshita, ibid 21, 462 (1906).
for example, H. E Ives and G. R Stilwell, Journal of the Optical Society of
J Cf
,
America, 28, 215 (1938).
I,
HISTORICAL SURVEY
with the principle of relativity.
How
can this result be understood on
the basis of the ether theory ?
In the first place, it should be observed that what
experiments
is
11
measured in these
not the phase velocity but the so-called ray velocity. In
is
Fizeau's original experiments, for example, a light signal is sent along a
certain path and back again, and the difference between the time of
departure and the time of return of the signal is measured. The velocity
is then determined as the ratio between the length of the path traversed
Now it is true that the velocity of the light signal
to
the
is equal
phase velocity c in the coordinate system 8 which is at
rest in the ether, but in the moving system 8' the velocity of the signal
will not be equal to the phase velocity r/ given by (24). This is plausible
and
this time interval.
one keeps in mind that a light signal represents a certain amount of
electromagnetic energy, and that energy, like mass, is a quantity which
if
conserved, so that a signal in many respects will resemble a material
particle. Therefore we should rather expect that the velocity of a light
is
moving system of coordinates S' is given by the equations
and (6) if in these equations u is put equal to c, the velocity of
signal in a
(3), (5),
light in the ether.
on the wave theory of light confirms this exa
shows that in a moving coordinate system
Such
treatment
pectation.
the ether acts like an anisotropic medium so that one has to distinguish
between the phase velocity and the ray velocity, which is identical with
the velocity of propagation of the light energy and is just given by (3).
To see this we shall apply the well-known Huyghens principle which for
all phenomena in the domain of geometrical optics is a consequence of
closer treatment based
Maxwell's equations. In accordance with this principle we obtain the
consecutive
wave surfaces
as the envelopes of elementary
from each point of a wave surface.
Let us consider the propagation of light
in a
waves starting
system of coordinates 8'
moving with the velocity v relative to the ether system S. In S' we thus
have an 'ether wind' of the velocity
v which will carry along the elewaves
in
the
same
as
sound
waves are carried by the
mentary
way
wind.
Fig. 2 gives a
diagram of successive positions of light waves in the
Let
the
surface a denote the position of a given wave front
system
at the time t. In order to construct the wave front a 1 at the time t-\-dt
8'.
point P on a as a starting-point of an elementary wave.
Because of the ether wind, this elementary wave will at the time / \-dt
obviously form a sphere E with centre at a point Q which lies at a distance
we regard every
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
12
v dt from the point
vector
PQ
is
P in the direction of the ether wind.
The infinitesimal
thus given by
PQ =
is
I,
-vdt.
(25)
Since the velocity of propagation of the elementary wave in the ether
= c dt. The wave front o is now
c, the sphere E has a radius QP^
FIG
2.
obtained as the envelope of all the elementary waves, i.e. the vector QPl
is perpendicular to a 1 at the point of contact Pv and in the limit as dt ->
QP
is
also perpendicular to
Thus, the vector
or.
of the phase-velocity vector, and
^.
where c
velocity
is
nc
by
is
QPl lies in the direction
we have consequently
c dt
en
(26)
dt,
the phase-velocity vector in the ether. In S' the phase
definition given
by
PA =
where the unit vector
c'dt
c'n' dt,
(27)
n' denotes the direction of the
Obviously
wave normal
in S'
(2g)
Since any infinitesimal part of a curved wave
surface can. be regarded as plane, the connexion between c' and c must
again be given by (24). This follows also directly from (27), (28), (26),
in accordance with (17).
and
(25) if
we note that
PB = QP =
l
while
BA
is
c dt,
equal to the projection of the vector
on the direction
n.
BP
PQ
dt
HISTORICAL SURVEY
I,
13
The relative direction of the ray, i.e. the direction of propagation of the
light
energy as estimated by an observer in
now given by the direc-
$', is
>
tion of the vector
PPV
and
we have
u' denotes the relative velocity of the ray
if
>
PP =
l
where
e' is
u' dt
u'e' dt,
(29)
a unit vector indicating the relative direction of the ray.
V
FIG. 3
Now
the vector
PP
is
the
sum
of the vectors
>
>
In the limit as dt ->
this gives,
u'
because of
PQ
and Ql\
- ->
(25), (26),
v.
and
(29),
(31)
In the absolute system the ray velocity is identical with the phase velocity
u
u
i.e.
ue
e
c,
so that (31) can be written
u'
en,
(32)
n',
v.
(33)
Thus we obtain the same addition theorem for ray velocities as for
particle velocities- u is the geometrical sum of u' and v. If & and &'
are the angles between the direction of the velocity vector v and the
absolute and relative ray directions, respectively, a consideration of
the triangle in Fig. 3 gives at once, since u -
f,
(34))
-v/c'
and
By
u' 2
+v 2 + 2vu' cos #' = u 2
solving this equation with respect to u'
{c
we
c2
get
-t; 2 +(v.e') 2}*-(v.e').
(35)
14
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
I,
A comparison between (35) and (24) shows that the relative ray velocity
in general
is
different
from the relative phase velocity, the difference
being of the second order in v/c. Only when the direction of the ray
(and the direction of the wave normal) is equal or opposite to the direcand c+v,
tion of v, are the two velocities identical and equal to
cv
respectively.
Now it is clear that the velocity measured by Fizeau's and Foucault's
methods is the ray velocity, but, since v also enters into (35), it should be
possible in principle to determine the absolute velocity of the earth on
the basis of these measurements. It is, however, easy to understand
variation in the velocity of light was ever observed. In these
experiments a light ray is sent along a known closed path and the time
why no
which the
light signal takes to travel along this
path
is
measured. In
make
the path as long as possible within a limited space, the
is
reflected
light ray
many times by suitably arranged mirrors. Let
the
be
traversed by the ray between the mirrors, and
distances
^i> ^25-"> ^
let the corresponding directions of the ray be given by the unit vectors
order to
e i>
2v>
e/
e l; then
we obviously have
since the ray describes a closed polygon. The time needed for the light
to traverse this closed path is then, according to (35),
t
^C
^{
If this expression
is
2_ v 2 +(v
expanded
e ; )2
On account of (36), the
mation we obtain
p_
in terms of the small quantity v/c
neglecting terms of order higher than the
t
37 \
(v e ;j'
.
we
get,
first,
first-order
term disappears and in
this approxi-
Thus, when terms of order higher than the first are neglected, the
measured time is the same as if the earth were at rest in the ether. A
determination of the absolute velocity v would consequently require a
measurement of quantities of at least second order. Fizeau's and
Foucault's methods, however, did not allow such a high accuracy, and it
is therefore understandable even on the basis of the ether theory that
*
the experimental results were in agreement with the principle of
I,
HISTORICAL SURVEY
15
was not until many years later that Michelson was able
method which also allowed the measurement of magnitudes
of the second order and thereby gave a final proof of the validity of the
relativity.
It
to develop a
We shall,
however, follow the historical trend of
development and return to a discussion of Michelson 's experiments in a
principle of relativity.
later section
12).
velocity of light in refractive media
we have only discussed the propagation of light in empty
Let
us now assume that the space is filled with an isotropic
space.
transparent substance with the index of refraction n. If the substance
7.
The
Hitherto
at rest relative to the world ether, the phase velocity in the absolute
system S is, according to Maxwell's phenomenological electrodynamics,
is
r 1==
where
e is
|,
the dielectric constant of the
(38)
medium and
/t
its
magnetic
permeability The phase velocity c\ relative to a moving system of coordinates is then, in analogy to (-4), given by
^-^-(n.v).
(39)
the refractive body is at rest in the absolute
the
8
But
body moves \\ith a velocity v, thus being at
system
suppose
rest in 8' what will then be the expression for the phase \ elocity in S' ?
This formula
is
valid
if
This problem was much discussed in the early days. The simplest
assumption is that (39) remains valid, i.e that the ether passes undisturbed through the moving body without being dragged along. Then
we have in ft' an ether wind with the velocity v, and we can now find
the ray velocity by means of Huyghens's principle in the same way as in
6. In equation (35) wo have simply to replace c by the phase velocity
ct
c/7i and thus get the following expression lor the ray velocity in S'
u'
--
[c\
-v*+(v.e')*}i-
(v.e').
(40)
Instead of assuming that the ether passes undisturbed through the
moving body, it has also been suggested that the ether is completely
dragged along by the body. According to this hypothesis, which was put
forward by Stokes, | we obviously get
u'
for, in this case,
c[
=2
(41)
there would be no ether wind in S'.
t G. G Stokos, Phil
134 (1880).
Mag
(3),
27, 9 (1845), Mathematical and Physical Papers,
1,
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
16
I,
A third possibility
is to assume that the ether is dragged along only
the
partly by
moving body, say with a velocity av, where the 'dragging
coefficient' a is a positive number smaller than 1 depending on the re-
fractive index n.
on the basis of the
This hypothesis was put forward by Fresnelf who,
gave the following expression for
elastic ether theory,
the dragging coefficient
=!--,.
On
(42)
this hypothesis the velocity of S' relative to the
(v/w
and, instead of
velocity
where
c*
we obtain
(39),
dragged ether
for the relative
is
phase
4=
c/n
is
()
4-^.
the phase velocity in a system of coordinates S*
Consequently the phase velocity in
accompanying the dragged ether.
the absolute system S is
Ci
since
= f+
c*+ a (v.n)
S is moving with
the velocity
cxv
(44)
.n)/l-l),
relative to S*.
In order to find the relative ray velocity u' by means of the method
outlined in 6, we must let the system of coordinates 8* take over the
role played by S in the former considerations. Since the ether wind in S'
2
has the velocity
(v/n ), we get, in analogy to (31),
u'
= C?-^,
n*
(45)
where the vector c* with the magnitude c* = c/n is the phase-velocity
vector in S*. The magnitude of the relative ray velocity we obtain in
2
v by c*
the same way from (35) by replacing c and
(v/n ),
c/n and
i.e.
respectively,
In the absolute system S, however, we have an ether wind with the
2
(1
velocity av
l/n )v. Therefore we have for the absolute ray
velocity u, in analogy to (45)
cf+av,
1
p^
1
J
n2
u
,*
'
| A. J. Fresnel,
and
2
{cf
(46),
-aV+a
(v.e)
}+a(v.e))
c
...
'
n'
Ann. de chim.
et
de phys. 9, 57 (1818).
<)
HISTORICAL SURVEY
I,
or,
neglecting terms of the second order in
addition theorem
v,
= - + a(v.e).
(48)
elimination of cj from (45) and (47)
By
17
==
we obtain again the simple
u >_|_ y
A comparison between the equations (43), (46), (44), and (47) shows
that the ray velocities are identical with the phase velocities if the direction of propagation of the light is the same as or opposite to the direction
of the velocity v. Without any calculation this follows immediately from
the fact that, in this case, the ether wind carries the elementary waves
in a direction parallel to the light beam. The considerations given in
this section are valid also for the case of in homogeneous bodies with a
continuously varying index of refraction. Only in this case the system
$*, which depends on the value of n, will be different at different points
of the substance.
8.
Hoek's and Fizeau's experiments
A measurement of the velocity of light in transparent substances seems
to offer a
new
the earth.
possibility for a determination of the absolute
An experiment of this kind was performed in
motion of
1868 by
Hoekf
who used an
interferometer arrangement of the type shown in Fig. 4.
monochromatic light ray from a source of light L is divided by a
(weakly silver-coated) glass plate P which is placed at an angle of 45
into a transmitted part 1 and a reflected part 2. The transmitted ray 1
by the mirrors Sl9 $2 $3 and
traverses a rectangular path
a
certain
fraction
of
the
P/S^/Sg/S^P; again
ray passes the plate P and
enters the telescope T. The reflected ray 2 traverses the same rectangle
is
reflected
in the opposite direction.
where it interferes with 1
filled
On its return to P it is partly
reflected into
Between S2 and $3 is inserted a tube of length I
with a substance with refractive index n (for instance water).
.
Even if the whole apparatus were at rest in the ether, such an arrangement would give rise to interference fringes in the telescope, since the
slope of the mirrors cannot possibly be adjusted so accurately that two
rays 1 and 2 which focus on the same point in the telescope have traversed
a path of exactly the same optical length. However, if the whole apparatus has a velocity v with respect to the ether, this will cause an extra
phase difference &F between the rays 1 and 2, which can be calculated
by means of (40), (41), or (46).
t
M. Hook, Archives Nderlandaises des Sciences Exactes
n
3595.60
et
Naturelles, 3, 180 (1868),
18
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
Let
us, for simplicity,
assume that the apparatus
is
set
up
I,
in such a
PSl and$2 S3
are parallel to the direction of motion v
of the apparatus relative to the world ether. The phase difference A
resulting from the absolute motion of the apparatus will then obviously
way
that the lines
be due to a difference in the times
and t 2 which the rays 1 and 2, respecA B and the corresponding
tively, require to traverse the tube of length
on the path connecting
distance
and /S\.
CD
On
the remaining paths
rays are completely equivalent so that no
contribution to the phase difference &F can arise from these parts.
DS1 S2 B and A$3 PC the two
-FIG
The values of the
to which the ether
is
no dragging at
is
all
>
4.
on the degree
2 will now depend
If there
refractive
substance.
the
dragged along by
quantities
we
and
get, according to (40),
cv
c/nv
c/n-{-v'
c+v'
where we have put the refractive index of air equal to 1. The phase
difference between the rays 2 and 1 in so far as it is due to the absolute
motion of the apparatus, is then
,
AJF
= v^-tj =
2lw
(c/n)
or, if
we
neglect terms of order higher than the
cv
first in v/c,
(50)
C
When the apparatus
is
with the absolute motion
at rest with respect to the earth, v is identical
of the earth. If, therefore, the apparatus were
rotated 180 around an axis perpendicular to the direction of motion of
AjF. Such a
the earth, the phase difference in question would be
rotation should thus cause a shift of the interference lines corresponding
HISTORICAL SURVEY
I,
19
Af
to a phase shift of 2&F and, since
according to (50) is a quantity of
be easily observable.
should
effect
this
in
order
the first
v/c,
The
result of
Hoek's experiments was, however, negative; no observ-
able shift of the interference lines could be detected after a rotation of the
apparatus. This result too
to-
agreement with the principle of
phenomena should be independent of
in complete
is
which
all
relativity according
the state of motion of the measuring instruments.
Since the accuracy of Hoek's experiments did not go beyond terms of
the first order, the negative result of this experiment was, however, not
a serious difficulty for the ether theory; it only showed that the equation
(40), based on the assumption that the ether is not dragged along by the
refractive body, could not be maintained. Similarly, Stokes 's hypothesis
is ruled out by the result of Hoek's experiment, for from (41) we would
getf
AF=-^
i;A/f1
(51)
On
the other hand, the formulae (43) and (45), corresponding to a
dragging coefficient (42), are seen to give an explanation of the result
obtained by Hock. In this case, we get, by means of (45) or (46),
AL1
]i
/
yj
\c/n
v/n
c-\-v
c/n-\-v/n
\I
v]
(52)
and if we neglect terms of order higher than the first, this quantity is zero.
immediately that only Fresnel 's value (42) for the dragging
coefficient oc gives a zero value for AF in first approximation. Hoek's
experiment can therefore be regarded as an experimental verification of
It
is
also seen
Fresnel's formulae for the velocity of light in
regards terms of first order.
moving
bodies, at least as
As early as 1851, Fizeau j had obtained the same result by measuring
the velocity of light in running water. The experimental arrangement
was very similar to the interferometer arrangement in Hoek's experi-
ment
The only
that the light rays 1 and 2 here
are passing through water on the path PSl as well as on the path S2 S^.
As indicated in the figure, the ray 1 is traversing the water in a direction
opposite to the direction of motion of the water, while the ray 2 has the
(see Fig. 5).
difference
same direction of motion as the water.
is
Now Fizeau compared the posi-
tion of the interference
fringes while the water
was at
rest in the tubes
Fizeau, C.R. 33, 349 (1851) A. A. Michelson and E. W. Morley, Amer. Journ. of
Science, 31, 377 (1886); H. Fizeau, Ann. d. Phys. und Chem., Erg. 3, 457 (1853).
t
20
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
I,
with that when a strong water current was sent through the tubes.
shift of the fringes could be observed.
marked
From Hoek's experiment we know
tive to the ether can
in our calculation
that the motion of the earth rela-
have an effect of second order only; we may therefore
make
the absolute system 8.
the assumption that the apparatus
The velocity vector v in
(48)
is
is
at rest in
then simply equal
\s
to the velocity of the water relative to the tube. Since e is parallel to v
on the paths
and CD, which are the only paths which give rise to a
AB
phase difference between
and
2,
we obtain from
(48) for this
phase
difference
where
water.
means the whole path through which the light ray travels in the
The shift in the position of the fringes observed by Fizeau was in
complete agreement with the phase difference given by
9.
(53).
Lorentz's theory of electrons
The experiments mentioned
in the preceding section
may be regarded
as a decisive experimental verification of Fresnel's formulae (42)-(48),
at least as regards all terms of first order. However, the derivation of
these formulae from the point of view of the primitive ether theory
meets with a serious difficulty when we keep in mind that the index of
on the frequency of the observed light.
Since the dragging coefficient a in (42) is a function of n, this would mean
that the dragging of the ether is not only dependent on the properties
of the moving body, but also on the frequency of the light. Strictly
refraction generally depends
HISTORICAL SURVEY
I,
21
a separate ether for each colour
speaking, one would have to introduce
the
of
light.
of course, an impossible assumption, and this difficulty imderboth Fresnel's mechanical ether theory and Maxwell's phenomeno-
This
lies
is,
which the ether represents the system of reference where
Maxwell's equations are valid. In fact, the dependence of the dragging
phenomena on the frequency makes it impossible to decide in which
logical theory in
system of reference the equations of Maxwell's electrodynamics are valid.
This difficulty is closely connected with the fact that the index of
refraction in this theory is constant and equal to (eju)* and thus does not
satisfactory
give any explanation of the dispersion phenomena.
explanation of the dispersion and of the dragging phenomena was given
by Lorentz f in his theory of electrons. According to this theory, the
by refractive substances, but stays constantly
the absolute system. The material
at rest in a certain system of inertia
bodies are assumed to be composed of atoms which contain a number of
positively and negatively charged electric particles. While the positive
particles contain practically the whole mass of the atom, the negative
particles, the electrons, are supposed to be very light. Under the influence
of the electromagnetic fields in a light wave they perform forced vibrations around their equilibrium positions. Therefore the electrons themselves will emit electromagnetic waves which interfere with the incident
wave in such a way that the effective velocity of propagation of the light
ether
is
not dragged at
medium
all
c/n instead of c.
According to this theory it is also clear that the coefficient n in general
depends on the position of the frequency of the incident wave relative to
in a
at rest
is
the proper frequencies of the electrons. Furthermore, Lorentz was able
show that a uniform motion of the refractive body modifies the waves
to
emitted by the vibrating electrons in such a way that the effective phase
velocity of the light in the moving body to a first approximation is given
by Fresnel's formulae (43) and (44).
Thus
as regards the propagation of light in refractive bodies Lorentz 's
electron theory gave, at least to a first approximation, the same results as
Fresnel's theory, avoiding, however, the serious objections which could
be raised against Fresnel's derivation of his formula. In one respect it
even gave a more precise formulation of Fresnel's formula (44). Since the
index of refraction n in dispersive media depends on the frequency v,
and since the frequencies on account of the Doppler effect are different
t
H. A. Lorentz, The Theory of Electrons, Leipzig, 1916. See also L. Rosenfeld, Theory
of Electrons,
Amsterdam,
1951.
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
22
I,
8', it must be specified which value for the frequency and thus
n should be inserted in (44). Now Lorentz was able to show that one
must insert the value n(v'), where v is the frequency in the system 8'
moving along with the refractive body, while n is the function of the
frequency which is valid for a body at rest in the ether.
We shall not go deeper into Lorentz 's theory since all the results mentioned above will be derived later in a much simpler way from the theory
of relativity (Chapter II). Here we shall confine ourselves to the remark
8 and in
in
for
that Fresnel's formulae (42)-(48) are a consequence of the electron theory
we neglect all terms of order higher than the first. In the following
if
we
formulae to show that, for all optical effects of
the first order, the ether theory in the form given it by Lorentz yields
results which are in agreement with the postulate of relativity.
section
10.
shall use these
Agreement between the ether theory and the principle
relativity as regards
all effects of
the first order.
of
Fermat's
principle
According to the principle of relativity the track of a light ray connecting two points which are fixed relative to the earth should be completely independent of the absolute motion of the earth. This must at
least be true approximately, otherwise it would be impossible to make
constant optical images of objects. If the passage of the light rays
through the lens systems of optical instruments were markedly dependent on the absolute motion of the earth, the image formation in such an
instrument would be time-dependent, an effect which has, however,
never been observed.
As was shown by Lorentz, f
this fact can easily be explained on the
basis of Lorentz 's electron theory when we assume that all terms of
the second order are too small to be measured. This result of Lorentz 's
the more remarkable as the relative ray velocity u' is, to a first
approximation, essentially dependent on the absolute velocity v. Neglecting all terms of higher than the first order in (46), we get
theory
is
u'
.=-
c/n
(v.e')/n
(54)
In order to construct the track of a ray on the basis of this expression
shall again consider Fig. 2 (p. 12). As in (29), we have
we
where
u' is
given by (54) and
e' is
relative ray through the point P.
u'e' dt,
a unit vector in the direction of the
But instead of
f See ref., p. 21.
(25)
and
(26)
we have
HISTORICAL SURVEY
10
I,
now QPl
= c dt/n and PQ =
v dt/n 2 where n
23
is
the index of refraction
at the place in the medium
the system $*, which is at rest in the 'dragged' ether, plays the same role
as the absolute system S in vacuo. Let us call two points P and Pl9
considered, for in the present consideration
which are lying on the same ray and on consecutive wave planes,
a and a v conjugated points. Then the construction of the light ray
obviously consists in a determination of the conjugated points on the
consecutive wave planes. Now consider two arbitrary points on a and <r 1
respectively, with the distance ds and let us form the quantity dsju'
,
',
where u' is given by
with the direction e' equal to the direction of the
(54),
and
connecting these points. If the two points are conjugated as
in Fig. 2, ds/u' is equal to dt, where dt is the time which the wave front
takes to travel from the position a to the position a v If the two points
line
are not conjugated as, for instance, P and E in Fig. 2, ds/u' will always
be larger than dt. For, in this case, we have dt
PR'/u' where PR' is
and
the distance from P to the point of intersection between the line
9
PR
wave E, and since 11 lies outside E, we have ds >> u dt.
and B be two fixed points in the refractive body. Consider
the elementary
Now let A
the integral
n
'"
u
A
along an arbitrary curve connecting the points A and B, where u' at
any point of the curve is the relative ray velocity (54) corresponding to
According to the above arguments the
then assume the lowest value when the curve coincides
the direction of the element
integral (55) will
ds.
with the light ray through the points A and B, because only in this case
elements ds of the curve connect conjugated points.
will all
The ray between two arbitrary fixed points A and B in the refractive
body is thus determined by the condition that the integral (55) is a minimum for the track of the ray, and since the integral is equal to the time
which the light ray needs to travel from A to J5, this theorem is identical
with Fermat's principle which is thus a consequence of Huyghens's
principle.
For the integrand
mation
in (55)
we
get,
u'
c/n
first
approxi-
c2
B
d*
as a
,v
(v.e')
=I
(54),
=n
(v.e')/n
B
Hence,
by means of
Llv.
\
(57)
ds),
'
24
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
10
I,
e' ds is an infinitesimal vector joining two consecutive
where ds
on
the
curve connecting A and B. The last term in (57) is conpoints
2
sequently equal to v/c times the projection of the curve on the direction
of v and this projection is the same for all curves connecting the fixed
points A and B.
Thus, for the purpose of variation
we can
replace (55)
/*-/V
A
A
This expression
is,
to
to travel from
by
<
5s
however, equal to the time which the ray would need
B if the refracting body were at rest in the ether. To
approximation the light track between two points in the moving
body is thus the same as if the body were at rest, in agreement with the
principle of relativity. If the path of the ray in a medium at rest is
mapped out by means of suitably arranged screens with small openings,
a
first
the ray will also pass through these openings
moving with constant velocity.
if
the whole apparatus
is
Hoek's experiment showed that the interference phenomenon occurring in his special experimental arrangement, at least in first approximation, was independent of the absolute motion of the earth. Nor has any
influence of the absolute motion of the earth ever been detected in the
numerous later interference experiments. These facts, which are in complete agreement with the postulate of relativity, can, however, as shown
by Lorentz,*)" easily be explained on the basis of the ether theory if we
may assume
that terms of the second order are below the accuracy of
the experiments.
Let us consider an arbitrary interference experiment where all parts
of the apparatus are at rest relative to the system of reference S' which
Such an experiment always involves
which start from the same point A and are brought to
follows the motion of the earth.
two rays
and
after having traversed different paths
interference at another point
to B. Now, according to (57), the times t l and t 2 which
I and II from
the rays
and
2 take to travel
from
B are
to
(59)
8
II
II
i(v.Jd
II
x
Seeref., p. 21.
HISTORICAL SURVEY
10
I,
25
where the integrals occurring in (59) should be taken along the two paths
I and II from A to B. Since these paths have common end-points, the last
terms in the equations (59) are equal and the difference in time is simply
given by
= - =
1
n
C n
-efc- (~ds.
Thus
same as
(60)
ii
apparatus were at rest in the
ether. Since the absolute velocity of the earth does not enter into (60) it is
obvious that the phase difference between the rays 1 and 2 at the point B,
which is obtained by multiplying the time difference by the frequency,
the time difference is the
if the
remains unchanged when the apparatus is rotated so as to give a different
position with respect to the direction of motion of the earth. Therefore
such a rotation (to a first approximation) will not cause any shift of the
interference fringes.
II.
The aberration
of light
As we have seen
in the preceding paragraph, the direction of a ray of
rate to a first approximation, independent of the absolute
at
any
motion of the light source and the observer. However, the direction of a
light ray depends essentially on the velocity of the light source relative
to the observer. This phenomenon, which is called aberration, was observed in 1727 by Bradley f who noticed that the stars seem to perform a
collective annual motion in the sky. This apparent motion is simply due
to the fact that the observed direction of a light ray coming from a star
light
is,
depends on the velocity of the earth relative to the star.
In order to find the magnitude of the aberration we consider a point
P' just outside the atmosphere of the earth, but in a fixed position relative to the system of reference S' following the earth. According to the
above considerations, the aberration depends only on the relative
velocity between the star and the observer, and we may therefore, for
the sake of simplicity, assume that the star is at rest in the absolute
system 8.
Now we
consider a light ray which comes from the star
and
passes through the point P'. The absolute direction of the ray thus
determines the direction in which the star would be observed if the earth
were at
while the relative direction of the ray determines the
apparent position of the star. Since the index of refraction n is equal
to 1 at the point P', the connexion between the absolute and the relative
rest,
direction of the ray is given by formula (34). Let 6 and 6' be the angles
between the direction of motion of the earth and the actual and the
t J. Bradley, Phil. Trans. 35, 637 (1728).
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
26
apparent directions to the star, respectively.
and #' = 77+0', and from (34) we get
tan0'=
We
have then &
11
I,
--.
(61)
costf+u/c
now
the path of the light ray from P' to the astronomer's telescope on the earth, at any rate in first approximation, is independent of
the state of motion of the earth, the light will not suffer any further
Since
aberration on this path. This
is
true even
if
the ray in the course of
its
path passes through a strongly refracting medium as, for example, when
the telescope is filled with water. Such an experiment was performed by
Airy (1871),f who showed that the magnitude of the aberration was not
changed by the presence of the water. The aberration formula
proved to be in complete agreement with observations.
12.
(61)
has
to
now
Michelson's experiment
As we have
seen, the results of all experiments discussed
up
were in agreement with the postulate of relativity however, the accuracy
,
of these measurements (with the exception of Ives's experiment which
was performed much later) was not good enough to allow the measure-
ment of terms of higher than the first order. To this approximation, however, Lorentz's electron theory, which is based on the concept of
absolute ether, was in agreement with the postulate of relativity.
an
motion of the earth relative to the ether should, according to Lorentz's
theory, influence the terms of the second order only. In order to obtain
a decisive experimental test of the postulate of relativity it was, therefore, of the utmost importance to devise an experimental arrangement
permitting a measurement of quantities of the second order.
This was accomplished in 1881 by Michelson, J who measured the
velocity of light by means of the interferometer arrangement outlined
in Fig.
6.
By means of a glass plate P a light beam from a light source L
divided into two rays, 1 and 2, perpendicular to one another. The
transmitted ray 1 is reflected by a mirror St back to P, where some of the
is
ray
is
reflected further into a telescope T. Analogously, the ray 2 is
and part of the ray goes through the
by a mirror S2 back to
reflected
plate and enters the telescope where
it interferes with the ray 1
the apparatus were at rest in the ether we should obviously
observe a set of interference fringes in the telescope.
glass
Even
if
t G. B Airy, Proc. Roy. Soc. London, A, 20, 35 (1871), 21, 121 (1873); Phil. Mag.
43, 310 (1872).
} A. A. Michelson, Amer. Journ. of Science (3), 22, 20 (1881); A. A. Michelson and
E. W. Morley, ibid. 34, 333 (1887).
HISTORICAL SURVEY
12
I,
27
Now let us assume that the apparatus is placed in such a way that the
P/Sf x is
path
and
let
parallel to the direction of
the paths
Fio.
and
time
first
length
I.
By means of (35)
6.
AF between the rays
which is due to the motion of the apparatus in the ether. For the
which the ray 1 takes to travel from P to St and back again we
then easy to calculate the phase difference
it is
1
motion of the earth in the ether,
PSl and P^2 have the same
obtain
I
21
c-\-v
,
I
<2)
tr/cr
on the way forth and back is parallel to the vector v. In the same
which ray 2 takes to travel from P to S2 and
t
2
back and, since in this case e' is perpendicular to v along the whole path,
since e'
way we obtain the time
we
get
by means of
(35)
/
2
2
2/(c
v 2 )~*.
(63)
Neglecting terms of order higher than the second, we
mentioned phase difference
When the apparatus is rotated through an angle
therefore get for the
of 90, so that the path
PS2 now becomes
Such
A-F.
parallel to v, the difference in phase will be
a rotation of the apparatus should therefore cause a shift of the
interference fringes corresponding to a change in phase of 2 A F.
When the distance between the interference fringes is used as unit of
length, the phase difference 2A-F gives directly the shift of the inter-
ference fringes
by such a rotation of the apparatus. In Michelson's
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
28
I,
12
so that one would expect a shift in the
experiment, 2AF was about
of the distance between the fringes.
of
of
about
the
position
fringes
,
Despite the fact that Michelson would have been able to detect with
certainty a shift a hundred times smaller, he could not find any effect
at
all.
Thus
for the first
time we are confronted with an experiment
is true with an accuracy
indicating that the principle of relativity
of at least second order, f
13.
The contraction hypothesis
The
result of Michelson 's experiment
meant a very
serious difficulty
for the ether hypothesis. Michelson himself tried to explain the absence
of the effect by assuming that the ether was carried along by the earth
during its motion round the sun. In this case there would be no ether wind
at the surface of the earth, but perhaps at high altitudes.
Michelson
therefore repeated his experiment on a high mountain, but still without
any detectable effect. The assumption that the ether should be com-
by the earth is also in conflict with all optical
and
with
Lorentz's
electron theory, according to which the
experience
pletely dragged along
dragging is only partial inside the refracting media.
In order to explain the absence of any effect due to the motion of the
earth in Michelson 's experiment, LorentzJ and FitzGerald independently put forward the hypothesis that any rigid body moving with
velocity v is contracted in its direction of motion, the relative contraction
being equal to
experiment
(1
(Fig.
The length of the path PSl in Michelson 's
would then not be I, but 1(1 v*/c 2 )*, while the
v 2 /c 2 )*.
6)
length of PS2 is unchanged, since PS2 is at right angles to the direction
of motion of the apparatus. For the time t l we then obtain, instead of (62),
*!
t
being given by
2/(c
-V)-i
29
(63). In this case the phase difference
(65)
&F becomes zero
in agreement with Michelson 's experiment.
According to this strange hypothesis, a stick which has the length 1
when it is perpendicular to the direction of motion of the earth should
obtain the shorter length
f Michelsori'.s results have boon confirmed later by several investigators. See, for
example, K. J. Kennedy, Ptoc. Nat. Acad. 12, 621 (1926), and K. K. Illmgworth, Phys.
Rev. 30, 092 (1927). Contrary to these results Miller obtained a small effect. See
D. C. Miller, Rev. Mod. Phys. 5, 203 (1933).
t H. A. Loreritz, Amst. Verh., Akad. v. Wet. 1, 74 (1892).
G. F. FitzGerald, see O. Lodge, London Transact. (A) 184, 727 (1893), in particular
p. 749.
I,
HISTORICAL SURVEY
13
when
it is
29
turned so as to be parallel to the direction of motion of the
on the earth it will never be possible directly to measure
earth. Certainly
this shortening, since all bodies, thus also the measuring sticks, are
observer at rest in the ether outside the
shortened at equal rates.
An
earth would, however, in principle be able to observe the shortening
and he would find the earth and all objects on the earth contracted in
the direction of motion of the earth.
The contraction hypothesis looks rather startling at first sight, but, as
by Lorentz, f it is impossible to escape from it as long as the
conception of an absolute unmovable ether is maintained. For, from
stressed
this view-point, the result of Michelson's
experiment can directly be
taken as a proof of the contraction with the same justification as the
shift of the interference fringes when certain parts of the apparatus are
heated is taken as a proof of a change in length of the heated parts.
In order to make the hypothesis somewhat more acceptable, Lorentz
made an attempt at explaining the contraction phenomenon on the basis
of the electron theory. He actually succeeded in giving a plausible
explanation of the formula (66). Assuming that the material bodies
up of electrical particles which are held together exclusively
by means of electric forces, he was able to show that the equilibrium positions of the electrical particles in such a purely electrical
are built
system are changed in agreement with
when the system as a whole
The difficulty was only that the
(66),
given a constant velocity in the ether.
presupposition that the particles are held together exclusively by electric
forces could scarcely be assumed to be satisfied in the real substances.
is
In particular it was difficult to imagine how the charge of a single electron
could be held together, unless strong attractive forces of non-electrical
nature were active inside the electron. If one therefore assumes that the
is valid also for a single electron, as was actually
assumed by Lorentz, this must be regarded as a pure hypothesis which
cannot be based on the principles of the electron theory alone. The
Lorentz contraction therefore seemed to be a basic and universal pheno-
contraction formula (66)
menon underlying
the general laws of nature.
14. Validity of the principle of relativity for all
physical pheno-
mena
Michelson's experiment was only the first of a long series of attempts
motion of the earth relative to the ether. These experi-
to determine the
ments include both optical and purely electromagnetic arrangements,
t See
ref., p. 21.
FOUNDATIONS OF SPECIAL THEORY OF RELATIVITY
30
and
I,
14
was completely negative. All phenomena
appeared to be independent of the motion of the earth. At the end it was
impossible to doubt that the principle of relativity is valid exactly, not only
in each case the result
for the mechanical
phenomena, but
also for all optical
magnetic phenomena.
We shall not here enter into a detailed discussion of
all
and
electro-
these experi-
ments, but confine ourselves to recalling Ives's experiment mentioned
in
5, which showed that the Doppler effect also to a second approxi-
mation depends only on the relative velocity of the source of light relative
This
to the observer, in agreement with the principle of relativity.
some of the other experiments mentioned
above cannot be explained on the basis of the ether theory, even if the
contraction hypothesis is added to it; for the formula (22) does not
fact as well as the results of
contain any quantity which has the dimension of a length. Consequently
a hypothesis regarding the contraction of distances alone will not change
the formula (22) into the formula
by Ives's experiment.
Next, Lorentzf investigated the problem of which hypotheses must be
(II.
90) verified
introduced beside the contraction hypothesis so as to
make all predictions
now
of the ether theory in accordance with the principle of relativity
verified
by the experiments. He found that
it
was necessary
in every
inertial system to use a special time, the so-called local time,
which
from the time in the absolute ether system. According to
the contraction hypothesis, the length of a metre stick depends on the
is
different
absolute velocity of the inertial system considered. Similarly, the rate of
clocks and therefore also the unit of time should, according to this new
hypothesis, depend on the state of motion of the inertia! system. If the
basic equations of the electron theory in a
moving system of
inertia
are written in terms of these local time and space variables, they assume
the same form in any system of inertia. All electromagnetic phenomena
should therefore appear to be independent of the state of motion of the
frame of reference. In
this
way
it
was
for
some time made
possible to
maintain the concept of an absolute ether by the introduction of new
hypotheses, until Einstein (1905) realized that the very foundations
of the ether theory were seriously shaken
mentioned experiments.
t See rof., p. 21.
by the
results of the above-
II
RELATIVISTIC KINEMATICS
15.
Simultaneity of events
attempts to find any influence of the motion of the earth on
mechanical, optical, and electromagnetic phenomena gave rise to the
THE
fruitless
conviction
among
physicists that the principle of relativity
was valid
for all physical phenomena. This obviously changes the whole basis of
our description of nature, for, as already mentioned in 2, the concept
of an absolute ether system loses its physical meaning as soon as the
universal validity of the principle of relativity is accepted. All physical
phenomena will then take the same course of development in any system
of inertia and one can never
system is the absolute one
by any physical experiment decide which
become completely
All systems of inertia then
equivalent and it must be required of a satisfactory theory that
systems of inertia are treated on the same footing.
all
Einstein was the first to formulate this new standpoint and to draw the
consequences of it in his fundamental paper of 1905.f We have already
mentioned one of these consequences in 2. Since the fundamental equaMaxwell's equations must hold now in any
tions of electrodynamics
system of
inertia, it follows that the velocity of
vacuo must have the same constant value
system of
inertia.
This
is,
propagation of light in
10 10 cm. /sec. in every
of course, in conflict with the kinematical
concepts derived from our usual experience and which are expressed in
the addition theorem (I. 3). Thus the new experiences gained from
extremely accurate experiments, which are expressed in the principle
of relativity, compel us to make a revision of these kinematical concepts
which by habit had obtained an a priori validity in the minds of physicists. Now Einstein could show that a closer analysis of the concept of
velocity, i e. a discussion of the methods by which a measurement of
velocities can actually be performed, opens up the possibility of an un-
ambiguous description of physical phenomena in accordance with the
principle of relativity.
Let us consider a light signal which travels in a straight line from a
point A to another point B in a given system of inertia. The velocity
with which the light signal has travelled from A to B is then defined as
the ratio between the distance from
t A. Einstein, Ann.
4, 411 (1907).
d.
A to B and the time which the light
Phys. 17, 891 (1905); Jahrb.
d. Radioaktivitat
und
Elektronik,
32
RELATIVISTIC KINEMATICS
needs to travel from
to B.
II,
15
The measurement of the distance does not
involve any difficulties, but on second thoughts it becomes clear that
the measurement of the difference in time between the emission of the
from A and its arrival at B is not so simple. If we imagine the
time of emission ^ to be read on a clock placed at A, while the time of
arrival / 2 is read on another clock at B, the difference 2
^ obtained in
light signal
only give the real time which the light has taken to travel
from A to B if the clocks at A and B are put right. This obviously
requires that the hands of both clocks simultaneously are in the same
position. But how can we make sure that two events occurring in two
this
way
will
different places are simultaneous ?
It is possible to think of various methods to synchronize the clocks at
and B.
can, for example, carry a third clock, which is set according
We
B and adjust the clock at B according to it,
or we can use a time signal which is sent from A to B. We shall start by
to the clock at A,
from
to
a consideration of the latter method which, in practice, has proved to be
the most accurate. Let us imagine that the time signal is emitted from A
when the clock at A
set the clock at
records zero. In everyday life one would then usually
to zero when the time signal arrives at B. This is, of
course, not quite correct since the time signal is propagated with a finite
velocity. If we wish to be quite accurate, the clock has on the arrival of
the time signal at
to be put to l/u, where I is the distance from A to B,
is the velocity of the time signal. In order to be able to make this
correction it is thus necessary to know the velocity of the time signal,
and u
but a measurement of a velocity presupposes, as shown above, that two
clocks in different places are synchronized, which was just the problem
solve. Similar observations are
which the time signal should help us to
true
when we set the
clocks at
A and B by transporting a third clock from
to B. In this case, the transported clock has to be corrected for the
influence which the transportation might possibly have had on the
motion of the clock, but to find experimentally this influence it is
obviously necessary in advance to dispose of two clocks in different
places of which we know that they are synchronized. Here again we are
moving in a circle.
All methods for the regulation of clocks meet with the same fundamental difficulty. The concept of simultaneity between two events in
different places obviously has no exact objective meaning at all, since we
cannot give any experimental method by which this simultaneity could
be ascertained. The same is therefore true also for the concept of velocity
As stressed by Einstein, we must first define what we understand by
.
15
II,
RELATIVISTIC KINEMATICS
33
As to definitions of concepts we are, however, to some
extent free and, as we shall see in the following, it is possible to use such a
definition of simultaneity that the velocity of light is constantly equal to
simultaneity.
c in all inertial
systems.
The relativity of simultaneity
Let us imagine that we are placed in an arbitrary inertial system /
and that we are provided with a large number of clocks (standard clocks)
16.
which show the same rate when placed at
rest at the
places in / where we
In order to synchronize the clocks
clocks shall be distributed at
all
same place. These
want to make time
we shall use light
a
we
know
about
the propagation
deal
since
experimentally good
signals
and
similar
more
accurate
Fizeau's
measurements
of light.
experiment
measurements.
show, for instance, that the time which a light ray takes to traverse a
closed polygon is equal to the ratio between the total length of the
polygon and the universal constant c occurring in Maxwell's equations.
This time can be measured by a single clock placed at a fixed point of the
polygon, independently of the definition of simultaneity, and all distances
may be measured by means of standard measuring-sticks at rest in /.
Now we choose an arbitrary point as regulating centre, and in
order to synchronize the clocks at different places in 1, a light signal is
at the start of the
emitted from O in all directions. Let the clock at
signal record the time
when this
signal arrives at an arbitrary point
is put to
-f Z /c, where / is the distance from O to P measured
with standard measuring-sticks at rest in 1 In this way all the clocks in
the inertial system / are set in a definite way. Two events occurring at
the clock
two arbitrary points P and Pl arc said to be simultaneous when the
clocks at P and Pl record the same time at the moment when the events
occur. Such a definition of simultaneity is completely justified if it can
be shown that it does not contain any 'inconsistencies. In this connexion two condition;-* must be satisiied, viz.
1.
A signal starting from O r seconds later than the regulation signal,
at the time -\~r, shall arrive at P r seconds later, i.e. when the
clock at P records the time +^o/ c T This condition means that
i.e.
~l~
the method of regulation of the clocks shall be independent of the
time when the regulation is made.
The method shall be independent of the choice of the point which is
taken as regulating centre.
The first condition is no doubt fulfilled, since all points in an inertial
2.
system are equivalent, so that two standard clocks which have the same
3595.60
RELATIVISTIC KINEMATICS
34
16
II,
when placed together at will also have the same rate when they
and P.
are installed at different points
As regards condition 2, we have only to show that a light signal emitted
from an arbitrary point l9 when the clock there records t l9 will arrive at
rate
P
another arbitrary point P when the clock at this point records the time
where
is
tt+l/c,
(I)
and P. In order to prove the equation
assume that the time of departure ^ of the
the distance between
(1) we may, for simplicity,
signal coincides with the arrival at
of the regulation signal from 0,
i.e.
from Px immediately after its arrival at P is sent on to
time of arrival 2 at
is, according to the experimental
mentioned above (Fizeau's experiment), given by
If the signal
the point 0,
results
its
since the light signal has actually traversed a triangular path of total
length Z x +Z-Ho ( see Fi g- 7 )
The clock at
When
P showed the time when the signal from Px arrived at P.
t
the regulation signal from
arrived at P, the clock was put to
back to
it would, according
to Fizeau's experiment, have arrived there when the clock at
recorded
the time / +2/ /c. Now the signal from l is, however, first sent down to
f
t Q -\-l Ql
c.
If this signal
had been
reflected
r
it
(o+^o/ c ) seconds
therefore arrives at
later.
when the
According to the above assumption
clock there records the time
.
(4)
RELATIVISTIC KINEMATICS
16
II,
This time
we now
is
identical with the time
given in
(3).
From
35
(3), (4),
and
(2)
obtain
is just the equation (1) which we wanted to prove.
Thus the regulation method used is independent of the point chosen as
regulating centre and in this way we have established a definite way of
This
recording successive events in the inertial system /. Any event occurring
at a point P at the moment when the standard ckrck at this place shows
simply said to occur at the time t. Then also the concept of
velocity assumes an exact meaning and, especially for the velocity of
light, we obviously obtain the value c in all directions. For a light signal
emitted from an arbitrary point Px at an arbitrary time ^ has just been
the time
is
shown
is
to arrive at another arbitrary point
the distance between
and v
P at the time ^-fZ/c, where
Let us now consider another arbitrary inertial system /'. Suppose that
in this system also we place a great number of standard clocks of identical
These clocks are distributed at the differand regulated in the same way as the clocks in / by means
construction to those used in
ent points of /'
/.
of light signals emitted from an arbitrary point 0' in I'. All distances in
/' are now supposed to be measured with standard measuring-sticks at
The measuring-sticks shall be of the same type as those used
which means that they have the same length when brought to rest
rest in /'.
in /,
Since, according to the principle of"relativity,
Fizeau's experiment gives the same result in /' as in /, it is clear that this
method of synchronizing the clocks in /' provides a consistent time
relative to each other.
description.
When the distances and the time differences are measured with clocks
and measuring-sticks in /', it then follows in the same way as in / that
the velocity of light relative to /' is also constant and equal to c in all
An event occurring at a given point P' of /' at the moment
clock at this point registers the time t' is said to occur at the
relative to /'. In general this time will be different from the time
directions.
when the
time
which the same event occurs relative to
Two
events occurring at
different points are now, of course, called simultaneous relative to /'
if they occur at the same time t in /'.
.at
/.
Thus the concept of simultaneity has lost its absolute meaning since
two events occurring simultaneously for observers in / generally will
not be simultaneous for observers in
two events occurring at two points
the velocity of light
is c
/'.
Let
and
B which are fixed in I.
us, for instance, consider
Since
in .all directions, the criterion for these events to
RELATIVISTIC KINEMATICS
36
be simultaneous relative to /
from
and
is
16
II,
obviously that two light signals emitted
the events occur shall meet in the
moment when
at the
of the line connecting A and B. A similar criterion for simultaneity is true also relative to /'. Now let the two events be simultaneous
relative to / and let us, for instance, imagine that the line connecting
centre
and B is parallel to the direction of the velocity v of /' relative to /.
Then consider the two points A and B' in /' which, at the moment when
the events occur, coincide with the points A and B. Simultaneously
(relative to /) the centre C' between A and B will coincide with C.
Since now C', just as A' and B', moves together with 1' with a velocity
v relative to
C' will not coincide with
/,
and
meet
light
signals from
meet
in C' and, according to the
events are not simultaneous
in C.
The
at the
above-mentioned
relative to 1'
moment when
the
light signals will thus not
criterion, the
two
The concept of simultaneity between two events in different space
points consequently has an exact meaning only in relation to a given
approximation where the velocity of light
can be regarded as infinitely great compared with all other velocities
which occur is it permissible to speak of an absolute simultaneity independent of the states of motion of the observers. Such an approximation
inertial system.
Only
in the
quite sufficient in daily life and in many cases also in physics, and this
explains the deep-rooted subjective belief in the existence of an absolute
is
time and in absolute simultaneity.
17.
The
special Lorentz transformation
In a given inertial system / an event which occurs at a point P at the
t can be characterized
by four figures, viz. the three coordinates
time
and the time parameter
specifying the point
called the space-time coordinates of the event.
These four figures are
If, for instance, we use a
t.
Cartesian system of coordinates in the inertial system
coordinates of the event are {x,
y, z, t}, where x
(x, y, z)
7,
the space-time
are the Cartesian
coordinates of the point P. The coordinates (x, y, z) are found by measuring the lengths of the projections of the coordinate vector x on the
Cartesian axes by means of standard measuring-sticks at rest in the
inertial
system
/,
while the time
is
read on the standard clock which
is
placed at rest at the point P.
In this
way a
definite
to the inertial system /.
system of space-time coordinates S is attached
When S is given, the frame of reference, i.e. the
/, is completely determined. On the other hand, different
of
systems
space-time coordinates S can, of course, be used in the same
inertial
system
II,
RELATIVISTIC KINEMATICS
17
37
/, for example, we may use polar coordinates instead
of Cartesian coordinates for the specification of the different points in
space. In the special theory of relativity we shall, however, always use
frame of reference
space-time coordinates of the above-mentioned kind, and thus need
not distinguish between the frame? of reference / and the coordinate
system 8. On the other hand, in the general theory of relativity it will
appear necessary to differentiate between the frame of reference and the
system of coordinates which is used for the fixation of the events occurring in the frame of reference.
s'
x
O'
FIG. 8.
If we consider another inertial system, an event will also in this
system
be specified by four space-time coordinates (x' y', z t') defining a spacetime system of coordinates S' The coordinates (jc',y', z', t') are found in
,
ti
way
by means of standard measuringand standard clocks now at rest in /'. Our primary task will be to
find the connexion between the space-time coordinates of the same
event in S and $', i.e. the transformation corresponding to the Galilean
the same
as the coordinates in
sticks
transformation
(I. 1)
Since any uniform
uniform also relative to S', the
in non-relativistic kinematics.
translatory motion relative to S
is
must obviously be linear functions of (x,y,z,t).
For convenience we shall assume that the Cartesian axes in S and 8'
are parallel to each other and that S is moving relative to S with
velocity v in the direction of the positive #-axis. Let us, moreover, assume
variables
(x' ,y'\z' ,2')
that the origin Q' of S' coincides with the origin
t
0.
of
at the time
RELATIVISTIC KINEMATICS
38
Now consider all the points in
y'
which form a plane
S'
= a' =
constant
These points
parallel to the z'z'-plane.
H, f 17
=a=
(5)
form a plane
will also
constant
(6)
The constants a' and a denote the distances between these planes and the #z-plane, and, since these distances
are measured by means of measuring-sticks in different states of motion,
in
parallel to the #z-plane.
the ratio
= a' fa
(7)
might turn out to be different from 1.
The ratio K can, however, only depend on the relative velocity v and
a simple relativity argument shows that K actually must be equal to 1
.
For,
if
we change the
z'-axes, neither
exchanged
a nor a'
roles,
and z-axes as well as on the x'- and
is changed, but now the two inertial systems have
signs on the
x-
S now moving with
the velocity v relative to S' in the
we may conclude, just as before,
direction of the positive #'-axis. Hence,
that
From
Q\
(8)
a/a
I
.
2
1 and, since the
however, that /c
positive
directions of the y- and the y'-axes are the same, a and a' must have the
same sign. Hence,
K
fl
a
(7)
and
(8) it follows,
(9)
These considerations show that an event occurring at a point with the
coordinate y in S will have a coordinate y' with respect to $', which is
V=
given by
do)
y.
In the same way we find that the z-coordinate is transformed according
to the equation
nLi
,
__
z.
In order to find the transformation equations for the two other space
we make use of the fact that a light signal in S and in
-
time coordinates
same velocity c. If the light
and 0' at the time t = t' = 0,
signal starts frqjn the coinciding points
the propagation of the spherical light wave is described in S by the
S'
is
propagated in
all
directions with the
x * +y * +z *- c *t*
equation
In S' the wave
is
and
put
(12)
analogously given by the equation
s'2
Now
s
*'
+y + 3'a_ c ya =
'2
=#
=*
+2/
/2
+z -c
+7/'
o.
(12')
2 2
J
+z -cV 2
(13)
/2
(13')
RELATIVISTIC KINEMATICS
17
II,
For any
zero,
s'
which makes s 2 equal
and since the connexion between
set of values of the variables (x, y,
must then
also
be zero,
and
(z',y',z',O
this is only possible
is linear,
when
to
(x,y,z,t)
2
proportional to s
is
K (v)s 2
i.e.
(14)
is
equation
(8)
we see at once that the constant K must
Thus (14) reduces to
1.
z, t)
a constant which can depend only on the relative velocity v.
the same relativity argument which was used in connexion with
where K
By
s'
s'
39
__
actually be equal to
nr;\
lc
>
S",
the quantity s defined by (13) is an invariant.
By means of (10) and (11), (15) can be written in the form
i.e.
x*-c*t*
= x'
-c 2
t'
(16)
and
Since (10) should be fulfilled for an arbitrary event,
and t alone. Therefore we can write
x'
t'
must be
linear functions of x
where the constants
fied for all
x and
]8,
y, 8
x'
t'
otx+8t,
(17)
have to be determined so that
(16) is satis-
t.
For the origin O' we have x' = 0. Thus we obtain from the
equation (17) for the motion of 0' relative to S
x
- _#/,
and, since the velocity of 0' relative to
J8
For the origin
first
is v, it
follows that
at?.
(18)
we have, however, x = 0. By introducing x
we get, after elimination of t, the equation
--
into
the equations (17)
x'
- j8/'/8
relative to S'. For
describing the motion of
relative to S' must, however, be
velocity of
(18) gives
ft
By means of
(18)
and
= _ 8v =
(19)
_^
symmetry reasons the
v,
wiich together with
ieg==a
equation (17)
(19)
can be written in the form
= oi(xvt),
f = yx +
x'
(20)
ott.
Introduction of these equations into (16) gives
(21)
RELATIVISTIC KINEMATICS
40
II,
17
Since this latter equation will be satified by all possible values of the
2
2
independent variables x and t, the coefficient of the variables x t and
,
on both
must be equal. This gives
three equations for the determination of the two quantities a and y. The
last two of these equations give immediately
xt,
respectively,
sides of the equation
(l-v
/c )-*
(22)
^-^/C2= ^VO=5^-
and
The remaining equation, expressing the
(23)
fact that the coefficients of x 2
on both sides of the equation are equal, is then identically fulfilled, which
shows that equation (19), i.e. the assumption that relative to >SY/ moves
with a velocity
From
v, is
in accordance with equation (16).
(10), (11), (20), (22),
and
(23)
we
finally get the following trans-
formation equations for the space-time coordinates of an arbitrary event
vt
=
(24)
y
vxfc
The
inverse relations which are obtained
(24)
with respect to the variables
x, y, z,
by solving the four equations
t
are
x'+vt'
They may be obtained from (24) by interchanging the primed and the
v.
unprimed variables and replacing v by
In 1 8) the quantity v was introduced as the velocity of 0' relative to S.
It follows from (24), however, that any fixed point P' in S with constant
values of coordinates x', y' z' moves with velocity v relative to S in the
direction of the #-axis. Analogously, we see from (24') that each fixed
v relative to AS" in the direction of
point P in S moves with velocity
v
therefore
denotes
The
the '-axis.
simply the relative velocity
quantity
of the two systems of inertia.
Lorentz was the first to introduce the transformation equations (24)
and (24') and, therefore, they are usually called Lorentz transformations. The derivation of these equations from the point of view of the
(
RELATIVISTIC KINEMATICS
17
II,
principle of relativity is, however,
41
due to Einstein. | In view of the special
position of the Cartesian axes in Fig. 8, we here speak of especial Lorentz
transformation. Under such a transformation the quantity s 2 defined
by
(13) is invariant.
If
Galilean transformation
18.
we put
goes over into the special
oo, (24)
(I. 2).
The most general Lorentz transformation
The transformation of the space-time coordinates
in the
more general
where the relative velocity of S' and S is not parallel to the
#-axis and where the rectangular coordinates in S and S' have arbitrary
orientations relative to each other, can obviously be obtained by means
of a suitable combination of spatial rotations of the axes in S' and 8
case,
2
together with a special Lorentz transformation (24). Since s is un2
changed by spatial rotations, s is thus also invariant by these. more
general Lorentz transformations.
Sometimes we shall need explicit expressions for the Lorentz transformations in the general case and it is convenient then to use the following vector representation. Let us, for a moment, again consider a special
Lorentz transformation corresponding to the orientation of the coordi8. We can now depict the space vectors
nates shown in Fig.
(x,y,z)
and
x'
(x',y',z')
of the space points of the event considered in S and $', respectively,
in one and the same vector space, (x, y, z) and (#', y' z') being regarded as
the components of the image vectors in & fixed system of coordinates in
',
this abstract three-dimensional vector space.
In the same vector space
the vector v, representing the velocity of the system S' relative to S,
is depicted as a vector with components (v, 0, 0). The special Lorentz
transformation (24) can then obviously be written as a relation between
the image vectors x, x', v and the variables t and t' of the form
where (v. x) = v x x-{-v v y-\-vz z\^ the scalar product of the image vectors
x and v. It is immediately seen that the three components of the vector
equation in (25) are identical with the first three equations
Introducing in the same manner the image vector v'
which represents the velocity of the systenr$ relative to
t Cf., for example, A. Emstoin, Ober
Braunschweig 1917.
die spezielle
und
S'
(24).
',
v, 0, 0)
the inverse
die allgemeine Relatwitatstheome,
RELATIVISTIC KINEMATICS
42
equations
can obviously be written in the form
(24')
18
II,
~-=
(25')
Since
v'
the inverse equations
- -v
(25') are in this case
(26)
obtained from (25) by inter-
and (x,) and replacing v by
v.
changing (x',')
of
the
Cartesian
axes
in
a
rotation
S
the
components of x and v
By
undergo an orthogonal transformation, and since the axes in the abstract
vector space are supposed to be fixed, this means that the rotation of the
(
'artesian axes in
x and
S induces a corresponding inverse rotation of the image
while the vectors x' and v' remain unchanged. In a
similar way, a rotation of the Cartesian axes in S' induces the corresponding inverse rotation of the image vectors x' and v'.
vectors
v,
Now let us first consider the case where the
Cartesian axes in
S and
S'
are subjected to the same rotations (starting from their position as shown
in Fig. 8). This means that the sets of variables (x,y,z) and (x ,y',z')
are subjected to orthogonal transformations with the same coefficients.
r
Then the image vectors
x, v, x', v' also suffer the same rotations and
between
these image vectors will still be given
therefore the relations
by (25), (25'), and (26). In this case we speak of a Lorentz transforma-
tion without rotation, since the angles (measured in $and S', respectively)
through which the Cartesian axes in S and S' should be turned in order
shown in Fig. 8 are the same, so that in a certain
sense the Cartesian axes in S and S' have the same orientation (cf.,
to obtain the orientation
however, the considerations in 19).
If ?' r v ir rs denote the components of the velocity of the system S'
relative to ft, and if we write y
v 2 /c 2 )-*, the vector equations (25)
(1
are only a short way of writing the four equations
,
-l)t^
y'
(y-i)v'*aA>H-{i +
]) VzVx
(y
r
=-
yv x x/c
x/v
(y-W
+(yl)vs v y y/v 2 +{l + (y
z
yv v y/c -~yvz z/C
l)vl/v*}z
(27)
vz yt
-\-yt
which thus represent a general Lorentz transformation without rotation.
On account of (26) the inverse equations (25') are again obtained from
(27)
by interchanging the variables
stituting
vx
~vz
Vy,
(x,y,z,t)
and
(x',y',z',t')
for (v x ,v y ,vs ), respectively.
and sub-
RELATIVISTIC KINEMATICS
18
II,
43
Proceeding now to the consideration of the case where the Cartesian
axes in S and S' do not have the same orientation, we must keep in mind
that the Cartesian axes in S and S' must be subjected to different rotations
in order to attain the orientation of the axes
shown in Fig.
equation (25) remains valid without change, the
to be replaced by
first
last
Dx'
where
is
x+v{(y-l)(x.v)/*;
-r
8.
While the
equation has
(28 o)
*}>
the rotation operator which transforms the image vector x'
Dx' corresponding to a Lorentz transformation without
into the vector
Thus X)" 1 represents the rotation of the Cartesian axes in S',
which would give these axes the same orientation (in the above-men-
rotation.
tioned sense) as the axes in S. Instead of the equation (26)
Dv'
we have
~v,
(29)
the components (v'x vy vz ) of the velocity of S relative to S' are in
vy
vz ). Multiplying (28 a) by the inverse
this case not equal to
x
1
rotation operator D" and applying (29), the Lorentz transformation
i.e.
(v
in this general case be written in the
may
=
=
X'
*'
form
3>-ix-v'{(y-l)(x.v)/-y*}
]
/'
y{*-(V.X)/C}
can also be interpreted as the rotation which has to be applied to the
S in order to obtain the same orientation of the axes in S and 8'.
axes in
The
relations inverse to (286) are therefore
= X>x'-v(-
=v
which
hand
also
can be proved directly by introducing
(28') into
the right-
and
(28) satisfy the
equation
which can be written in the vector form
(x.x)
If
side of (286).
It is easily seen that the equations (25)
(15)
(28')
)>
we put
oo,
(25)
c 2 t 2 == (x'.x')
cV 2
(30)
becomes the general Galilean transformation
(I. 1).
Until
time
now we have assumed
that the origins
and 0' coincide at the
and, accordingly, the Lorentz transformations are
transformations
of the space-time coordinates. We shall
homogeneous
now abandon this assumption and consider a displacement of the origin
of the space and time coordinates in S'. This means that we have to
t
t'
replace
(x' 9 y',z',t
in (24), (25),
and
(28)
by x'-x'Q) y'-y*, z'~4 *'-&
RELATIVISTIC KINEMATICS
44
respectively,
where
x'Q
y'Q
z ()
are constants.
t'
II,
18
By an inhomogeneous
Lorentz transformation of this type the quantity s 2 will not be invariant
any more. However, if we consider two events with the coordinates
rt
//!,
2^
/!
and
j\2
7/0,
As
between
-=
z2
>
respectively, the differences
^2>
A//
.^-.r,,,
//r~//2>
he coordinates of the two events will also be transformed for
an inhomogeneous Lorentz transformation according to the equations
(25), (28) since
the constants
y'Qj z'
.rj,,
t'
(}
will
ences are formed. Therefore the quantity
A,s
A.r 2
2
|
A?/
+ A;:
disappear when the differdefined by the equation
>2
A.s
2
~ r 2 A/ 2
(31)
be invariant by an arbitrary mhomogeneous Lorentz transformation.
According to the principle of relativity all physical phenomena have
will
the same course ol development in all systems of inertia and it must be
required that, in the theoretical description of the phenomena, all inertial
systems are treated on the same footing,
e.
the fundamental equations
of physics must have the same form in every inertial system. In other
words the fundamental equations must be form-invariant or covariant
under Lorentz transformations. This requirement, which is the formal
expression of the principle of relativity, has proved to be very useful in
the development of new theories.
As we
requirement of form-invariance is autoior Maxwell's fundamental equations of electro-
shall see below, this
matically fulfilled
dynamics in vacuo. On the other hand, Newton 's fundamental equations
of mechanics do not satisfy this requirement, since these equations, as
shown on pp. 3-4, are covariant under Galilean transformations.
Newtonian mechanics, therefore, is valid only in the approximation where
Lorentz transformations and (! all lean transformations can be regarded
as identical, i.e. where all the velocities occurring are small compared
with the velocity of light. But for all mechanical phenomena, where
velocities of the same order of magnitude as the velocity of light are
involved, the Newtonian equations must be replaced by Einstein's
rclativistic equations of mechanics which are covariant under Lorentz
transformations
19.
(cf.
Chapter
III).
Contraction of bodies in motion
From
the Lorentz transformation (24)
we can now draw
certain con-
clusions regarding the intercomparison of measuring-sticks
in the
systems 8 and
relative to 8'
and is
and clocks
Consider a measuring-rod which is at rest
placed parallel to the o;'-axis (Fig. 8). The end-points
8'.
II,
RELATIVISTIC KINEMATICS
19
45
of the rod therefore have constant coordinates x^ and x\ and the length
of the rod in S'
(its
rest length) is
70
i
~/ ~~ ~'
^
2
*!
According to the first equation (24) the motion of the two end-points
relative to S is given by the equations
Now it is natural to define the length
of the rod relative to
S as the differ-
ence between simultaneous coordinate values of the end-points; by
simultaneity in this connexion we understand simultaneity relative to S.
From
(32)
I
we then obtain
x 2 (t)~^(t)
(xi-XiKI-vifc*)*
Z(l-t'
/c
(33)
)*,
which is independent of t. On the other hand, since the systems S and S'
are completely equivalent, a metre stick at rest on the .r-axis of S which
has the length 1 in 8 will have a length / relative to /S', which again is
given by
(33).
A metre stick which is placed perpendicular to the a:-axis will, however,
according to (24), have the same length in S as in 8' We may therefore
.
quite generally say that a body which moves with a velocity v relative
to an arbitrary inertial system 8 is contracted in the direction of its
motion according to the equation (33), while the transverse dimensions
are independent of the motion of the body. If F is the rest volume of the
body, i.e. the volume measured in an inertial system following the body
in its motion, its volume V in 8 is thus given by
V
The equation
(33)
is
--
V Q (}-v*/c^-.
identical with the
(34)
Lorentz equation
(I.
66).
However, as regards the physical interpretation, there is a difference in
principle between the two equations. In (I. 66) Z was the length of the
metre stick at rest in the ether while I was the length of the stick in
motion with a velocity v relative to the ether. Thus, according to the
original point of view of Lorentz, the metre stick is attributed an absolute
length which is independent of the state of motion of the observer. In
equation (33), however, P is the rest length of the metre stick, i.e. the
length measured in an inertial system following the stick, and I is
the length measured in an arbitrary inertial system relative to which the
stick has the velocity
v. Therefore, according to relativistic conceptions,
the notion of the length of a stick has an unambiguous meaning only
in relation to a given inertial frame, this length being different for the
RELATIVISTIC KINEMATICS
46
II,
19
This means, however, that the concept of
has
lost
its
absolute
meaning. We can only speak of an absolute
length
length in the approximation where the velocity of light can be regarded
different systems of inertia.
as infinitely large.
In spite of the fact that the concept of simultaneity enters into the
above definition of the length of a moving rod, equation (33), as pointed
out by Einstein, can in principle be verified by experiment without the
use of clocks. Let us consider two rods
and 2 with the same rest
l
S
the velocities v and
in
with
1
the
#-axis
v,
length
moving along
on
to
a
stick
the
of
Since
(33) depends
according
length
respectively.
the square of the velocity only,
must have the same constant
l and
2
length I relative to 8, which means that they must coincide at a certain
time t. In other words, the coincidences of the two ends of l with the
events relative to S. Let these
ends of
2 must be simultaneous
coincidences occur at the points A and B in S. Then a subsequent
measurement of the distance A B with a standard metre stick in S gives
the magnitude of I.
Even if it is, of course, impossible to perform such an experiment in
practice with an accuracy sufficiently high to verify equation (33), this
consideration shows that the Lorentz contraction given by (33) is a real
by experiment. It expresses, however, not
so much a quality of the moving stick itself as rather a reciprocal relation
effect observable in principle
between measuring-sticks in motion relative to each other. In this conit is natural to ask for the cause of the contraction. According to
the principle of relativity, the answer must be that such a question is
just as delusive as if, after the discovery of the law of inertia, the question
were put why a body left to itself will continue to move straight forward
with uniform velocity. While such a question was well justified in
Aristotelian physics it must be rejected as meaningless after Galileo's
discovery. According to Galilean and Newtonian mechanics only the
deviations from uniform translatory motions require a cause.
While Lorentz attempted to explain the contraction phenomenon on
nexion
the basis of the electron theory, Einstein's deduction of equation (33),
based on the principle of relativity alone, shows that the contraction
phenomenon
is
of a
much more fundamental
character.
Instead of
considering the contraction to be a phenomenon which has to be explained on the basis of an atomistic theory of material bodies, it should
rather be regarded as something elementary which cannot be traced back
to simpler phenomena. Actually it represents a requirement which must
be
satisfied
by any atomistic
theory, viz. the requirement of covariance
RFLATIVISTIC KINEMATICS
19
II,
47
If, on the other hand, this requirement
the contraction of a macroscopic body in motion can obviously
under Lorentz transformations.
is fulfilled,
be deduced from the theory of the atomic structure of the body.
Before leaving this section we shall consider the somewhat more
general case where the connexion between space-time coordinates is
given by a Lorentz transformation without rotation. Let x'x and X 2 be
the coordinate vectors of two fixed points Pi and P 2 in the system of
coordinates S'. The straight line connecting these points represents a
= x'2 xi in S'. At the time t the points Pi and P 2 will
have coordinate vectors x l and X 2 in S which are obtained from the
first equation (25) by putting x = x x x' = xi and x = X 2 x' = x 2
fixed vector r'
respectively.
o/ovi
-v*c*)~*
'
where
is
Then by subtraction of these equations we obtain
X2
")
1})
-.2
o c? \
-,'
(35)
x
'
xx
the vector connecting two simultaneous positions of Pi and P 2 in S.
TH+TJL is decomposed into its components r\\ and r parallel and
If r
perpendicular to v, respectively, and analogously
also be written
which shows that
it is
r'
which can be
From
r[,+r'_L, (35)
can
only the parallel component which suffers a
Lorentz contraction. The inverse relation to (35)
r
r'+v^-^{(l
v 2 /c*)*
is
(35')
1},
by inserting (35') in the right-hand side of (35).
follows that a fixed vector in S' parallel to the #'-axis is
verified
(35') it
in general not parallel to the #-axis as judged by an observer in S. Even
in the case of a Lorentz transformation without rotation the Cartesian
axes in S' will thus from the point of view of an observer in S generally
not be parallel to the axes in 8. Hence if we state that the Cartesian
axes have the same orientation in a Lorentz transformation without
way described on p. 42. For
=
r
in
S'
and
which
ri
2
satisfy the condition (ri r 2 )
we can very well have (r l r 2 ^ 0. Therefore the rectangular axes of
rotation this should be understood in the
two fixed vectors
coordinates in
will generally
not even be perpendicular to each other
when looked upon from the system S. For this reason it was necessary to
depict the vectors x and x' in the neutral abstract vector space introduced on
p. 41.
RELATIVISTIC KINEMATICS
48
20
II,
The retardation of moving clocks. The clock paradox
Now consider a standard clock C' which is placed at rest in S at a point
on the x'-axis with the coordinate x'
x\ (Fig. 8). When the clock C'
20.
records the time
at that
moment
tion (24'),
ti
Somewhat
later,
other clock in
By
standard clock in S which C' is passing by
a time ^ given by the Lorentz trarisforma-
== f the
lt
/'
will record
'
'
Y(t 1+vx 1/c 2)
when
$ which
(y
(1_ V 2/ C2) -J).
C" records the time
t'
t'
2,
coincides with an-
it
records the time
subtraction of these equations
we obtain
(36)
i.e.
a clock which
is
moving with the velocity
v relative to
will
be slow
compared with the clocks in S.
When we keep in mind that the systems S and S' are equivalent it is
obvious that a clock at rest in 8 similarly will lag behind the clocks
inS'.
Since the Lorentz transformation from which this result
is
deduced
is
based upon the method of synchronizing the clocks in S and /S', discussed in
16, one could think that the retardation of moving clocks
described by (36) was only apparent. However, just as in the case of
the Lorentz contraction, it can be shown that equation (36) contains a
statement regarding the rate of moving clocks which in principle may be
verified by experiment. Consider two standard clocks Cl and C2 placed
together at the origin of an arbitrary inertial frame of reference. At the
time
the clock
the velocity
C2
set in
is
At the time
v.
and, according to
(36), it will
uniform motion along the #-axis with
p it has reached a point P on the x-axis
record the time
p (lv
/c )*.
Immediately
after arriving at P, C2 is sent back to
with the velocity
v. It arrives
at
when the clock Cl records the time t = ^ = 2t p Since, according to (36), the rate of C2 is independent of the sign of v, C2 will on its
arrival at O record the time 2 ~ 2tp (l
v 2 jc 2 )*. The clocks C^ and C2
.
which at the
return of
C2
start of
,
C2
record times
e(l uation
The
difference
between
both showed the time
and
and
2,
zero, will thus, after the
respectively, connected
(l-t^/c'K.
t
may now
(37)
in principle be
directly by a comparison of the readings of C^ and
whole process.
by the
measured
C2 before and after the
RELATIVISTIC KINEMATICS
20
II,
49
his first paper on the theory of relativity,
in the past
of
the
theory gives rise to a paradox which
consequence
has played an important role in the discussions on the consistency ot the
of reference E which follows
theory. Suppose that we introduce a frame
to P and back again. Since, now,
the clock C2 during its motion from
the motion of C relative to R is quite analogous to the motion of
As mentioned by Einstein in
this
one would think that an observer in R would find that
the clock C\ is slower than Cz in contradiction with (37). This argument
is wrong, however, since equation (36) is valid only in an inertial
which during
system and therefore it is not applicable to the system R
C2 relative to
/S,
the change of velocity of C\ from v to
fixed stars.
is
accelerated relative to the
in R instead of
The question which equation should be used
cannot be answered in the special theory of relativity which only
allows treatment of the physical phenomena in frames of reference in
uniform motion. This discussion clearly shows the desirability of an
(36)
extension of the special theory of relativity to a general theory which
allows the use of systems of coordinates in arbitrary motion. (For the
98, p. 258 )
final solution of the clock paradox, see Chapter VIII,
a uniform
with
Now let us again consider a standard clock moving
#. The time recorded by the
u relative to an inertia!
system
velocity
to
moving clock itself is called the proper time r of the clock. According
the increase in proper time
(36) we have the following relation between
dr and the increase dt ol the time in S.
dr=
This equation
is
now assumed
is
the
clock where
(l-u*/c*)*
to be valid also for
momentary
the rate of the clock,
any lime
is the
the system in
It should
clocks from
and
to
moving
Hence we assume
arbitrarily
an inertial system has no influence
that the increase in the
proper time of the clock at
that of the standard clocks in the rest system
the clock is momentarily at rest.
same as
which
an
velocity of the clock.
that the acceleration of the clock relative
on
(38)
dt.
&,
i.e.
also be possible to deduce the retardation of moving
the fundamental laws of mechanics governing the running
now
it
of the clockwork. But, just as in the case of the Lorentz contraction,
an
as
elementary
is more adequate to regard the retardation phenomenon
of the principle of relativity.
which is a direct
consequence
phenomenon
ot
If we took Newtonian mechanics as a basis for the calculation
be
would
clocks
of
moving
the working of the clock, no retardation
found since the time in Newtonian fundamental equations is an invariant
the Newtonian equations
parameter (cf. I. 1 6), but this just shows that
,
3595.60
RELATIVISTIC KINEMATICS
50
II,
20
are not accurate in the region where (1
v 2 /c 2 )* differs appreciably from
1.
If, on the other hand, we use the exact relativistic equations of
mechanics in the description of the working of the clocks
III
(cf.
Chapters
and VI), the retardation effect must of course follow as a consequence
of these equations.
In view of the fact that an arbitrary physical system can be used as a
clock, we see that any physical system w hich is moving relative to a
r
system of inertia must have a slower course of development than the
at rest. Consider, for instance, a radioactive process. The
same system
mean
life r of the radioactive substance,
thus be larger than the mean life r
From (36) we obtain immediately
will
(1
^'
/c
when moving with a velocity v,
when the substance is at rest.
)~M.
(39)
In general, v is so small relative to c that we need not discriminate
between r and r. In recent years, however, very rapidly moving radiothe mesons have been observed in cosmic radiation
active systems
where the factor (1 v 2 /c 2 )~ can be of the order of magnitude of 100 or
more. Therefore equation (39) has been o( essential importance in the
J
interpretation of the phenomena connected with the decay of the mesons.
may also use a radiating atom as clock, the number of light waves
emitted per second being a measure of the rate of the atomic clock. If
We
the proper frequency of the atom, i.e. the frequency of the emitted
light measured in the system of inertia /S in which the atom is at rest,
v
is
number
of waves emitted per unit time in this system is just v.
In the system S relative to which the atom is moving with the velocity v,
2
the number of waves emitted per unit time will then be v(l
v*/c )*, for,
the
S corresponds to the time
If now the moving atom
$.
(1
system
has no radial velocity relative to an observer in S, this number will be
equal to the frequency v found by the observer in S, the number of
emitted waves being equal to the number of waves arriving per time unit.
Consequently we have
40
v = ^^^2^2)^0
when the radial velocity *is zero. This is, for example, the case when
the atom moves with the velocity v along a circle whose centre is the
observer or when the direction of the light arriving at the observer is
according to
interval
AT
(36),
a unit time interval in
v 2 /c 2 )* in the rest
perpendicular to the direction of motion of the atom.
According to the theory of relativity we must therefore expect a shift
in frequency also for perpendicularly incident light, viz. a shift towards
smaller frequencies, in contradistinction to the non-relativistic Doppler
RELATIVISTIC KINEMATICS
20
II,
51
formula (I. 14). This red shift of the spectral lines, the so-called 'transverse' Doppler effect to which we shall come back later ( 25, p. 62), is
a direct consequence of the retardation of moving clocks described by
and any experiment which permits an experimental verification
is therefore simultaneously an experimental proof of
(36),
of this effect
formula
21.
(36).
Transformation
of particle velocities
Let us again consider the two systems of inertia S and S' (Fig. 8) whose
space-time coordinates are connected by (24) and (24'). The motion of
an arbitrarily moving particle will then in S be described by a set of
equations
In S' the same motion
which
may
described
is
x'
y'
x'(t'),
y(()>
(41)
by functions
= y'(t'),
z'
(41')
z'(t'),
be obtained from the functions (41) by means of the Lorentz
The momentary velocity of the particle relative
transformations (24)
to S'
is
defined
by
,,-
,
.
Analogously, the corresponding quantities in
(42)
are given
Idx du dz\
-
'
"
(43)
'
(?
now
Differentiation of the special Lorentz transformation (24)
dx'
dt'
y(dxvdt),
e\\
/ A
dt' dt)
by
dy'
-^
dz'
dy,
dz
\
}
y(dt-v dx/c
gives
.
4
>
v"*
*/
/ 4
-\
from which we immediately obtain
'
1lx
^x
'
,
11
M'v\ I
2-
r-
'
)
,
Uz
'M'z\
^ /^
/
.
;
(45)
These equations for the transformation of the velocity reduce to the
ordinary transformation equations
(I.
4) in
the limit as
relations inverse to (45) are obtained in the usual
way by
the primed and the unprimed variables and substituting
c -> oo.
The
interchanging
v for
v.
RELATIVISTIC KINEMATICS
52
II,
If we choose the 2-axis so that u (and thus also u')
the z-axis, (45) can be written in the form
Q
U COS I/
i?
and
'
Qf
UVCQS&/C
where
t*/
OXIJL
perpendicular to
is
i/
JL
t/
lUVCOSl
'
are the angles between the #-axes and the vectors
i?
21
u and
r
,
respectively.
From
we obtain
these equations
tan T?'
'
Sm
^-~- -
(I. 5)
From
(I. 6).
formula
and thus
If u,
For
_^/c
2)i
---
we
(48)
get from (45) the
for velocities
^-~ ?;
__
calculation the
(i^ u 2/ c 2^i_ v 2f c ^}
u' -- c it gives also
From
we obtain by an elementary
theorem
'
relativistic generalization of the equations
also u' are parallel to the #-axis,
rHativistic addition
"
(47)
__.^/ c 2) (l
(J
(46)
* ~~
2
~~ 2v CQS &lu
1?2 8ln2 #A'
^ v2 / u
These equations are the
and
directly
__
u'-\-v
c.
follows directly that no systems
since the equations (24) as well as
the expressions for the Lorentz contraction and the retardation of clocks
would become imaginary in this case. But it can be shown, furthermore,
the Lorentz transformations
of inertia
y/
>S
can exist for which v
that particles
(or,
more
>
it
c,
generally, signals) cannot
move with
a velocity
any inertial system, since this would lead to
absurd results. Let us assume for a moment that we were able to emit
signals with a velocity greater than the velocity of light. At the time
-t
t' --- 0, where the two
systems of coordinates 8 and 8' in Fig. 8
coincide, we could then send a signal from the common origins 0, 0'
greater than
relative to
along the negative u/-axis with a constant velocity u'
At the time t\
0, this- signal would arrive at a point
>
'
1
a -axis
this
with the coordinate
event in 8
are,
Immediately after
x'
----
according to
its arrival in
iit\.
>
relative to
AS".
P on the negative
The space-time coordinates of
(24'),
P the signal is supposed to be sent back
II,
to
RELATIVISTIC KINEMATICS
21
with a velocity
w>
53
The motion of the
c relative to S.
signal
is
thus described by the equation
w(t-tj+xp
of S at a time t 2 which
from
this equation
by putting x
>
u'
we could obtain that
i.e.
w^
c*/v,
1
is
obtained
thus at the time
0,
yf^lu'v/ct+W
t^Xpliv
If we now choose u' and w so that
/
(51)
This signal will arrive at the origin
v)/}.
(52)
--
v/c
(53)'
-l
0,
(54)
O records a number
that, at the return of the signal to O, the clock at
smaller than that recorded by the same clock at the moment of
departure of the signal. Obviously this is impossible, and therefore we
can infer that in nature no signals can exist which move with a velocity
which
is
greater than the velocity of light relative to any system of inertia. This
represents a general statement regarding the fundamental laws of
]n particular, as we shall see later (Chap. Ill, 29, p. 76), a
material particle according to relativistic mechanics can never reach
nature.
velocities larger
than
r.
By differentiation of (25) and (25') we obtain the following transformation equations for velocities in the case of Lorentz transformations
without rotation
_
l-(v.u)/c
= (l_^c 2)iuM-I-l-?;c
These equations are reduced to
From (55) we deduce
V.U/c
in agreement with (48).
(1
When
(I.
w'^
)(L
u' is perpendicular to v,
u
and when u
is
parallel to
22. Successive
the limit as
3) in
2
)*
i.e.
= v + (l
v 2 /r 2 )i(l-M 2 /r, 2 )i
(1
for u'.v
2
i;
v we get back
c -> oo.
0, (55')
(56)
reduces to
/c )iu'
(57)
to the formula (49).
Lorentz transformations.
The Thomas pre-
cession
Let us now consider three
moves with the
velocity
incrtial
systems S,
S',
and S" of which S'
moves with the
relative to 8, while S"
RELATIVISTIC KINEMATICS
54
velocity u' relative to S'.
(x,t) in S and (x',f) in S'
II,
22
The connexion between the coordinates
then given by a (generally inhomogeneous) Lorentz transformation. In the same way, the connexion
between (x',f) and the coordinates (x" ,t") in S" is represented by a
Lorentz transformation.
By
is
elimination of the four variables (x',f)
between these eight equations we obtain the connexion between (x, t)
and (x", t") and, for physical reasons, this relation must also be a Lorentz
transformation. Mathematically this is expressed by the statement that
the Lorentz transformations form a group. If the origins in 8 and 8'
coincide at the time / ~~ t'
0, and if the same is the case with S' and
8" at the time
t"
t'
-= 0,
the origins in
8 and S"
will of course coincide
This shows that the homogeneous Lorentz transformations form a subgroup. It is also clear that the pure spatial rota-
at the time
t"
0.
tions of the Cartesian axes without
any change of the system of reference
form a subgroup.
In non-relativistic kinematics the Galilean transformations without
rotation of the Cartesian axes also form a subgroup. This is, however,
not the case in relativistic kinematics, for if we combine two Lorentz
transformations without rotation the resultant Lorentz transformation
will in general
correspond to a change of orientation of the Cartesian
8 to 8' be given by the Lorentz transforma-
axes. Let the transition from
tion (25), while the transition from 8' to 8" is described by the equation
obtained from (25) after replacing (x,,v) by (x',',u') and (x',') by
(x", t"). By elimination of x', t' we then obtain a Lorentz transformation
of the type of (286), viz.
X"
- D-IX-W"(*^[
where the operator
1"[
in general is different
-},
(58)
from the unity operator.
w is the velocity of the system 8" relative to S and w" is the velocity of
S
relative to S". Since the transformations
from S to S' and from S' to
8" were Lorentz transformations without rotation, the velocity of S'
relative to S" is equal to
u' while the velocity of 8 relative to S' is
to
v.
We
now
obtain
the
of S" relative to S from (55'),
equal
velocity
identifying u with
in this formula,
__
1
From
i.e.
the same equation
we
'
+ (U'.V)/C*
get the velocity
w"
of
relative to S"
''
by
KELATIVISTIC KINEMATICS
22
II,
replacing
v and
u'
u'
by
and
v, respectively; thus
--
__ 1
According to
(29)
we have
55
we have
'
+ (!'. V)/C
'
in our case
Dw" = ~w,
(60)
(59) and (59') shows that the rotation operator D
from the unity operator. Only if u' is parallel to
is
different
general
say equal to kv, we get from (59) and (59')
and a comparison of
in
v,
w = ~w =
i.e.
in this case
l+k
>
and the combined Lorentz transformation
is
also
without rotation.
Let us now consider the case where the transition from S' to S" is an
infinitesimal transformation,
i.e.
where
u' is infinitesimal.
Neglecting
terms of higher than first order in u', the transformation from S' to
S" is reduced to
2
x"
t"
x'-uY,
(61)
J'-(u'.x')/c
all
and
for
w and w" we obtain from (59) and (59')
W
,.
_/v +
'-v<X'?l>'
(62)
By substituting the expressions (25) for x' and t' in (61) we obtain
an equation which after, a simple, but lengthy, calculation may be written
in the form (58) with
dv
=w
Thus, we have
Dx =
The rotation operator I) thus represents an infinitesimal rotation around
the direction of the vector ft, the angle of rotation being equal to the
magnitude
|ft|
of the vector
ft.
With
I)
given by (64) and
and w"
RELATIVISTIC KINEMATICS
56
given by (62) it is easily verified that the equation (60)
the approximation considered.
Let us
now
some way
consider a point compass,
or other defines a direction. It
22
IT,
is
satisfied to
a material particle which in
may be assumed that a classical
e.
electron with spin represents such a point compass. Tf the velocity of
the particle relative to 8 is v
v(/), and if w e put dv
\(t) dt in (64),
the systems 8' and 8" in the above-mentioned considerations will be
momentary rest systems of inertia for the particle at the times t and
Since the transition from 8' to 8" represents an
t-\ dt, respectively.
infinitesimal Lorentz transformation without rotation, it is natural to
assume that the direction shown by the compass at the time
the same orientation relative to the axes in 8" as
it
-\-dt
has
has at the time
relative to the axes in 8' provided that the forces on the particle
,
do not
exert any torque on the compass.
Now, if we put dv - v dt, the rotation vector i defined by (64) represents the rotation which has to be applied to the axes in 8 at the time
order to give
t-}-dt in
them the same
orientation as have the axes in #".
Since, furthermore, the direction of the
system
to
is
compass relative to the rest
means that the direction of the compass relative
turned through an angle corresponding to the rotation vector SI.
is
constant, this
In other words, the compass performs a precession relative to
8 with
the
velocity of precession
w=
where v
r/v/rf/ is
obtain to a
first
--^K
-^
- l}(v X v),
(65)
the acceleration of the point compass. When?'
'ewe
approximation
This precession phenomenon was studied for the
Thomas
first
time by Thomasf
and
is
23.
Transformation of the characteristics of a wave according
therefore called the
precession.
to
the theory of relativity
Let us again consider a plane wave with the wave normal n in the
.ry-plane of a system of coordinates 8 and with the frequency v and the
velocity w relative to 8. It is described by one or several wave functions
of the form
T
t L.
=-,
A<**
W. Thomas,
TCOS
Phil
^Pl
,,--
May
(7),
3,
J?
,
(1927).
(67)
RELATIVISTIC KINEMATICS
23
II,
where a
is
the angle- between the
57
wave normal n and the
x-axis.
In a
system of coordinates S' moving in the direction of the ^-axis with the
velocity v relative to S (cf. Fig. 1), the wave is described by corresponding
functions obtained from (67) when replacing the quantities occurring in
the phase by the corresponding quantities measured in the system of
From
exactly the same argument as that used in
follows that the phase must be an invariant, i.e. the equation
coordinates S'.
it
must be
--
# cos tx -)-?/ sin <x\
__.^
---
(,
_-r
-- -.^'
,/,/
cos
cx'
valid for all points in space
?/'
sin a'\
w^
\t
and at
all
(bS)
I
times.
Now
using the
connexion between the space-time coordinates in A and S' given by the
Lorentz transformations (24') we can eliminate the variables .r, //, t in
this
equation and get
~~~
v cos OL/W
~~
cos
v sin
vw/c~ ~vx'
ex
y
,
w
an equation which must be
This
isnrx
----w y
satisfied for all values of the
/',
.r',
//.
is
I--VCO&OL/W
sn
W
v'
cos
OL
From
these equations
sn
W
y'',
(70)
w(\
rw/c
?;
2
)
/c
)-
we obtain
.,
COS
ex
(71)
VWJC*
wvconoi
2vw cos oc
The
.*;',
<x
i/(cos
w'
(<>9)
(v.n)/w
(\
independent
possible only if the coefficients of /',
on
both
sides
are
equal. Therefore we must have
respectively,
variables
v 2 w*
~~
v* sin 2 a
inverse relations are obtained in the usual
way by interchanging
the primed and the unprimed quantities and replacing v by
v. In the
limit as c -> oo, the equations (70), (71), and (72) reduce to the nonrelativistic
formulae
(I.
14), (I. 16),
and
(I.
19).
comparison of the transformation equations (46) and (47) for the
KKLATIVISTIC KINEMATICS
58
23
II,
and direction of a material particle with the formulae (71) and
shows
that (46) and (47) become (71) and (72), respectively,
(72)
when we put u -- c 2 /w, u' -- c 2 /w'. In other words, the velocity u of a
particle and its direction n are transformed in the same manner as the
c 2 /u
corresponding quantities for a wave with the phase velocity w
and direction n. In his wave theory of elementary particles de Broglief
made use of this circumstance by attributing to a particle with the
velocity ?/ and the direction of motion n a plane wave with the same
c 2 /u, a procedure
direction of propagation n and the phase velocity w
velocity
which thus
If the particle velocity
relativistically invariant.
is
corresponding wave velocity is w =
c, which shows that direction
and velocity of such a particle are transformed in the same way as are
direction and velocity of a plane light wave in vacuum, but this only holds
the
for this particular value of the velocity.
The ray
24.
??,
velocity in
moving bodies
Consider a homogeneous isotropic medium, with the refractive index
at rest in the system of coordinates S' (Fig. 8). Relative to S the
medium moves with
the velocity v. In the rest system S', Maxwell's
phenomenological equations of electrodynamics in dielectric bodies are
valid, and, according to the principle of relativity, this must be true for
any constant
r/
elocity of xS relative to the fixed stars.
The phase velocity
of light relative to $' is, therefore, w'
c/n in all directions. In this
system of reference the ray velocity must, however, be equal to the phase
velocity, since the elementary waves which, according to Huyghens's
determine the ray velocity are spherical waves with the con-
principle,
stant velocity
J
This
is
not
w
so,
---
/r.
c/n.
rt
(73)
however, in the system of coordinates S. Let us conpropagation of an elementary wave which, at the
is emitted from the common origin 0, O' of S and 8'.
sider, for instance, the
time
In
r/
>S
--
- 0,
the propagation of the
wave
x' 2 +y' 2 +z'*
where w'
is
given by
(73).
we obtain from (74) the
elementary wave in S:
is
described by the equation
w'H' 2
0,
By means of the Lorentz transformation
(74)
(24)
following equation for the propagation of the
t L. do Brogho, Th&so,
Pans
1924.
RELATIVISTIC KINEMATICS
II,
24
An
elementary wave which
is
emitted at the time
59
from a point
(:r
T/ O)
in the #?/-plane of the system S will thus, at the time J-f A, have a curve
of intersection with the o;?/-plane which is described by the equation
My; *o,yo) We
c
(*-*o-a to)*/b+(y-y Q )*-bw'*
shall here only deal
>
>
and
>
with the case n
>6>
1, i e.
w/
<
A* 2
c.
0.
In
this case
(76)
and the curve of intersection consequently
\
FKS.
9.
-\-a A, y Q and with the semiA, respectively. The elementary waves are thus
dragged along by the medium with the velocity a, and simultaneously
is
an
ellipse
axes bw'
with
A and
its
centre at the point (x
(}
b^w'
the waves are contracted in the direction of motion in the proportion
b^.
Now
let us again consider a plane wave with a normal n which lies in
the #?/-plane and makes an angle a. with the #-axis. The connexion
between a and the direction of the normal in 8' is then given by (71) or
by the inverse equation
2
Let
cr
is
wave plane in S whose
given by the equation
X Q cos oc +2/
ellipse
P:
(x
sin a
~ C
E with its centre at the point
elementary wave
(77)
cos a' -\-vw' /c 2
in Fig. 9 be a
xy -plane
The
(l--?; /r )ismrx'
tana
line of intersection
constant.
with the
(78)
Q: (x Q -{-a A, y Q ) represents the
is emitted from the point
at the time t-\-At, which
at the time
t.
It
is
given by the equation
(76).
The wave plane
RELATIVISTIC KINEMATICS
60
24
II,
t-\ A is represented by the line or v which is obtained as the
of
the manifold of ellipses (76) which are obtained by varying
envelope
the parameters (x Q9 y Q ) in accordance with (78).
at the time
The
and magnitude of the ray velocity are thus given by
direction
PPV where Pl with coordinates (#i,?/i)
touches the envelope a v We have obviously
the line
the point at which
is
y.
PP ^
Since P
l
A/,
(79)
is the limiting point of intersection
where u is the ray velocity.
l
of two adjacent ellipses of the manifold of curves (76), we see that the
coordinates of P besides satisfying the equation (76) must satisfy the
equation obtained by varying :r ?/ in (76) in accordance with (78), i.e. the
l
equation
-
or
----
sin a
(x
,r
ex
a A/)sina
XQ
From the three equations
of
cos
and y Q the equation
(76), (78),
b(y
and
~-
we
(80)
for the envelope
.rcostt-f-ysinrc
)cosa.
?/
<7
\-\a-\- w'{b(b-\
(80)
obtain by elimination
in the
form
tan 2 t\)] ]costt Af.
J
(81)
The last term in (81 gives the distance PA between the two wave planes
which must be equal to w A/, where w is the phase velocity. Therefore
we obtain
)
(82)
6, and a it is easily
inserting the expressions (75) and (77) for
is identical with the inverse of the equation (72).
According to (79) we have
By
verified that (82)
11
X I~~ X Q
*-^Ar'
__
V \-~lh
^-~AT~
/oo\
(83)
x lt y ~ y l is a solution of the equations (76) and
and, since r
obtain by a simple calculation
fc*
When inserting the expressions
that the ray velocity u'
that (84) is reduced to
is
(75)
equal to
--
The equations
(85) are identical
and
(80),
we
tan raw/
and remembering
the phase velocity w' in $', we find
(77) for a, 6, a
--'-with the relations inverse to
(45),
which
KELATIVISTIC KINEMATICS
24
II,
shows that the ray velocity
61
(just as in the absolute ether theory, cf.
transformed in the same way as the velocity of a material
(I.
When
& and &'
a.' denote the
particle.
angles between the x-axes and
33))
is
the direction of the ray measured in 8 and $', respectively, we have
therefore also for the ray velocities the equations (46) and (47) connecting
the quantities u, u'
&,
and &' By solving
.
with respect
(47)
twit,
we obtain
(86)
Here
and e
^'
is
(87)
c/w,
a unit vector in the direction of the ray,
v.e
In vacuum,
i.e.
for
I,
i.e.
i'cos#.
u'
(88)
u and
r,
are consequently
equal to c, and the formulae (46) and (47) for the ray velocity become
identical with the equations (71) and (72) for the phase velocity. According to the theory of relativity the ray velocity in vacuum is thus identical
,
with the phase velocity in any inert ial system. In the absolute ether
theory this was the case only in the absolute system. This difference is
due to the fact that the elementary waves in vacuum according to the
theory of relativity are spherical waves with a fixed centre in any system
of inertia. (For w ~ c we get from (75) a
and b -- .) We shall see
r
---
later (Chapters V, VII) that the ray velocity
which the electromagnetic energy
is
is
equal to the velocity with
flowing in an electromagnetic wa\
e.
and
The energy current density is, however, gi\ en by Poynting's vector,
wa\ e in vacuum it is found that Poynting's vector lies in the
direction of the wave normal in all systems of inertia.
This identity of ray velocity and phase velocity applies only to the
vacuum. In a refractive medium we have in general to distinguish
between the two velocities; only when the ray is parallel to the direction of motion of the medium, both the ray velocity and the phase
velocity are, according to (85) or (86) and (72), given by the same formula,
for a plane
ViZ
"
= &*?.,
lv/cn
(89)'
^
where the plus or the minus signs should be taken depending on whether
the ray travels in the same direction as the medium or opposite to it.
RELATIVISTIC KINEMATICS
62
II,
25
The Doppler effect, the aberration of light, and the dragging
phenomenon according to the theory of relativity
25.
The
relativistic
formula for the Doppler effect
is
obtained from (70)
identifying the inertial systems 8 and 8' with the rest systems of
the observer and of the light source, respectively. The frequency v is
then the frequency measured by the observer, and v is equal to the
by
c and the
proper frequency v of the light source. Now, in vacuum, w
direction of the ray is equal to the direction of the wave normal. Thus,
we obtain
l_Cv eWr
<>
LiirJ/r
//I
/
For
(90)
\
/
2 //2\
/
the velocity of the light source relative to the observer and
gives the direction of the light in the system of the observer.
where v
e
is
(v. e)
lar to the
when the direction of the observed light is perpendicudirection of motion of the light source, we get the formula for
0,
e.
the transverse Doppler effect v -- v(l v 2 /c 2 )*, which, as mentioned
in
20, is a direct expression for the retardation of moving clocks. As
the formula (90) has been experimentally verified with
high accuracy by Ives,f who measured the frequency of the light emitted
mentioned
in
5,
from rapidly moving ions.
The formula (46) connecting the directions of a
light ray in
two inertial
systems 8 and 8' leads immediately to the relativistic aberration
formula, when we take 8 and 8' to denote the systems of coordinates in
which the fixed stars and the earth, respectively, are at rest. At a point
c and, when introP' outside the atmosphere of the earth we have u
# TT and 6' = &' IT between the direction of
motion of the earth and the actual and apparent directions to the star,
ducing the angles 6
we obtain
the relativistic aberration formula
COS0 + V/C
from the equation (I. 61) of the ether theory by quantities
of second order only, a deviation which is negligible compared with the
present accuracy of the measurements.
It deviates
Since the atmosphere of the earth is at rest relative to 8', it follows at
once from the principle of relativity that the ray during its travel from
the point P' to the telescope does not undergo any further aberration.
According to the theory of relativity this should even hold exactly,
while Lorentz's absolute theory of electrons gave this result to a first
approximation only
(cf.
10).
t See ref., p. 10.
II,
RELATIVISTIC KINEMATICS
25
Another
63
essential difference in the relativistic treatment
is
that here
the aberration also appears in the direction of the phase velocity. While,
in the absolute theory, the wave normal according to (I. 28) has the same
direction in
S and in S',
tion of the
wave
the relativistic equation (71) for the transformanormal in vacuo is identical with the transformation
equation for the direction of the ray (46).
Neglecting all terms of higher than first order in v we obtain from equation (86) for the ray velocity in a medium moving with constant velocity
v relative to an arbitrary inertial system 8 the simple formula
u
where a
u '+v.e(l
u' 2 /c 2 ) =- c/r*+o:(v.e),
(92)
1/n* is Fresnel's dragging coefficient. Equation (92) is,
in this approximation, identical with Fresnel's equation (I. 48) for the
1
velocity of light in the 'absolute' system of coordinates. This formula
thus follows directly from the principle of relativity without any hypotheses regarding the atomic structure of the medium. It should be
stressed, however, that Fresnel's theory
is
true to a
first
approximation
In any other system Fresnel's equations
approximation. According to the theory of
only in the 'absolute' system.
are not valid even to a
relativity
we
first
have, for example, u'
c/n in the rest system of the
medium whereas
u' according to Fresnel's theory is given by the more
formula
(I. 46).
complicated
Equation (92) has been verified not only by Fizeau's experiments
discussed in 8, but also with high accuracy by Zeeman.f In these latter
experiments the velocity of light in a rapidly moving quartz rod was
measured. Indeed Zeeman's measurements were so accurate that it
was necessary to take into account the
effect which, according to
Lorentz's electron theory, should occur in a dispersive medium (cf. 9).
Since n
n(v) in such substances is dependent on v, and since v on
account of the Doppler effect depends on the system of coordinates, it
must be specified which value for v, and thus for n, should be introduced
into equation (92). It follows from the derivation of (86) and (92) given
above that we have in (92) to use the value of n corresponding to the
frequency v in the rest system of the medium. When we neglect all quantities of order higher than the first, the connexion between v and v,
according to
(70), is
given by
v'v
-=
-vn(v.e)lc,
(93)
the difference between the directions of the ray and of the wave normal
t P. Zeernan, Amst. Versl. 23, 245 (1914); 24, 18 (1915).
RELATIVISTIC KINEMATICS
64
being of second order in
Hence,
v.
n(v)
this
n(v)
n2
dn nv(v
dv
n(v) means the value of n
formula in (92) we obtain
where n
25
II,
e)
for the frequency
v.
By
using
(94)
This formula was in complete agreement with Zeeman's measurements.
The reflection of light by moving mirrors and the refraction of light at
the transition between moving media can be treated in a similar way
on the basis of the theory of relativity. In a system of inertia 8' in which
',
the mirror or the
medium
ray by a mirror and
at rest, the usual law for reflection of a light
the well-known law of refraction at the transition
between two media are
is
valid.
The corresponding laws
in the system
are then directly obtained by applying the transformation equations
(46) and (47) for the ray velocity. Experiments on reflection by moving
mirrors have been performed
is
by Sagnacf and others. The
result of these
in accordance with the theory.
experiments
In all these experiments, only effects of first order could be measured,
since the velocities with which the mirrors and the refractive media could
be moved were so small compared with the velocity of light that the terms
of second order in equation (86) were negligible compared with the
accuracy of the measurements. In this connexion it is instructive to
consider a slightly generalized Michelson experiment in which the whole
apparatus is filled with a refractive medium. From the principle of relativity
it is
clear that also in this case
ference fringes will occur
p. 27).
This result
is
no displacement of the
inter-
when the apparatus is rotated (cf. Fig. 6,
when we calculate the times t[ and z
also obtained
t'
which the two rays take to traverse the paths PS^ P and P*Sf2 P in the
system of inertia 8' in which the apparatus is at rest for in this system the
velocity of light is the same, viz. c/n in all directions and therefore t[ = 2'2
Now it must be required, of course, that the same zero result is ob;
when the
from the point of view of an observer
in a system 8, relative to which the whole apparatus has a velocity v.
The corresponding time-intervals t t and 2 measured by clocks in S,
may now be obtained from the formula (86) and it is easily seen that
again ^ = 2 The time which the ray 2 needs to travel from P to
tained
effect is treated
So
is
obviously \t 2
During
this
time the apparatus has moved over
t G. Sagnac, C.R. 157, 708, 1410 (1913), Joutn. de Phys. (5), 4, 177 (1914).
RKLATIVISTIC KINEMATICS
25
IT,
65
a distance \vt z in S, and the ray has traversed a length %ut 2 where the
is determined by (86) or (47). In Fig. 10 these distances
,
ray velocity u
are
shown
as
P and S
PP* and
PA!?*,
after the time
P* and xS* denoting the positions occupied
From the figure we see that the angle &
by
2
between the ray and the direction of v is given by
cos$ = v/u.
Substituting this in (47)
to
n2
we
obtain,
by solving this equation with respect
_. 9V
i/ 2
(95)
By means
$2
of the Pythagorean theorem applied
to the triangle
from Fig. 10
PP*8* we
obtain, moreover,
J? 2
(96)
15
(u-
V-)*
where / is the distance between the plate P
and the mirror 82 From (96) and (95) we then
get for the time t 2 which the ray 2 needs to
travel from P to 8 2 and back to P
.
2L
Proceeding to the calculation of the time
which the ray 1 needs to travel from P to S1
>
FIG. 10.
and back to P, we shall here use for the velocity
u the expressions (89), since the ray 1 all the time moves parallel to the
direction of motion of the apparatus. If we denote the time which this
ray needs to travel from P toSl by tf, the mirror $x will have moved a
distance vt during this time and the total distance which the ray has
travelled relative to 8 is equal to l-\-vtf, where
I
is
the distance between
(l~v
/c
(98)
)*
and 8l measured
in S.
Therefore
we have
l+v/cn
v/cn)
or
(99)
The time
simply^
by
t{
which the ray needs to travel from Sl back to S
replacing v
by
?;,
i.e.
l
3595.60
n(l
v/cn)
is
obtained
RELATIVISTIC KINEMATICS
66
The
total time
II,
25
then
is
'--'f^-i^v-
<
100 >
Thus t l is equal to t 2 as was to be required.
Even if this result only shows that the theory is consistent, it also
allows us to make a choice between Fresnel's formula (I. 47) and the
relativistic equation (86) which differ from each other in the terms of
,
second order.
performed, we
If we use (I. 47) instead of (86) in the calculations just
find for the times t l and t 2 the expressions
which shows that
that the distance
formula
for
PS
n
is
we would
obtain
y-
even
if
we assume
contracted according to the Lorentz contraction
(98).
In the same
way
as the negative result of the usual Michelson experias an experimental verification of Lorentz 's
ment can be regarded
(98), a negative result of the corresponding experiment, where
the apparatus is filled with a strongly refractive medium, would mean
a verification of the relativistic formula (86) also as regards terms of
formula
higher order. The same considerations can be applied to Hoek's experiment discussed in 8.
Summarizing, it can be stated that relativistic kinematics in all details
is in agreement with the experimental results. It allows a simple explanation of all dragging phenomena without any ad hoc hypotheses, and it
gives a formula for the Doppler effect which, in contradistinction to the
formula of the ether theory, is in accordance with the experimental
results.
Ill
RELATIVISTIC MECHANICS
Momentum and mass
of a particle
As mentioned in the conclusion of 18, it is necessary to change Newton 's
fundamental equations of mechanics in order to bring them into accordance with the principle of relativity In the domain, where all velocities
26.
compared with c, relativistic mechanics will, however, go over
into Newtonian mechanics. It is therefore natural to assume that such
fundamental concepts of Newtonian mechanics as momentum and mass
of a material particle also have a meaning in relativistic mechanics.
Therefore, to a material particle moving with the velocity u relative
are small
to a system of inertia S,
tional to
we
shall assign a
momentum
vector p propor-
mu.
(1)
The proportionality factor m is called the mass of the particle. To make
room for the above-mentioned generalization of mechanics we shall not,
is a constant, but we make the
a
that
universal
is
function, f(u), of the magnitude u
|u
assumption
of the velocity vector, thus
however, assume beforehand that
m=
m(u)=f(u).
(2)
If the velocity of the particle relative to another system of inertia S'
is u', the momentum and mass of the particle relative to S' must be
g iven
p'=
where
is
the same function of to'
m'u',
(3)
m' = m'(u') = f(u')
(4)
as m is of u. This follows from the principle of
relativity according to which all systems of inertia have to be treated
on the same footing, so that any relation between physical quantities shall
be form-invariant.
It will now be our
primary task to determine the function /. As we
determined when we require that the
shall see, this function is uniquely
theorem of conservation of momentum shall hold in any system of
Let S and S' be two systems of inertia with the relative velocity
v, and consider a collision between two identical particles 1 and 2 which
before the collision have the velocities u x and U 2 relative to S. The corresponding velocities relative to S' are then determined by (II. 55).
inertia. |
t G. N. Lewis
and R.
C.
Tolman, Phil. Mag. 18, -510
(1909).
RKLATIVISTIC MECHANICS
68
Let us now choose the velocities before the
of
it
55)
(II.
26.
collision so that
- -u r
u;
By means
III,
(6)
follows then at once that also
-u
--=
u;
(6)
2.
u l and U 2
and u[ and u'2 relative to 8'. We shall in particular consider
in which the final and initial velocities of particle 1 have
After the collision the particles will have other velocities,
relative to 8,
a collision
opposite directions,
i.e.
= -au
Ui
where
is
(7)
t,
a positive number. For symmetry reasons we must then also
have
u2
-=
--u 2
(8)
with the same proportionality factor a as in (7), for, according to (5),
the motion of particle 2 relative to an observer in 8' must be the same as
the motion of particle
(8)^
0'8
which, according to
an observer
relative to
(II. 55),
Assuming that the functional
/(%)i+/K)2
(9)
-u
2.
relation (2)
is
momentum
in
the same before and
gives
=/(*i)Ui+/(w 2 )u 2
(10)
assume that the velocity of particle
collision is perpendicular to v,
(5), (7),
and
before the
(11)
(9) this involves
K
and
i.e.
(i^.vJ-O.
According to
and
involves
after the collision, conservation of
us, furthermore,
(5), (7),
=--fli,
o; =.
Let
From
in 8.
(u t .v)
---
v)
--
(12)
(u^.v) -= 0.
(13)
For the transformation of the velocities u.> and U 2 we can therefore use
the simple formula (II. 57). Thus we get
u2
and
=r
u 2 (l~r 2 /c 2 )H v
7*5
-=
(U 2 .u 2 )
--
- v-u^l-^/c
+v
/c
?/f(l
?
the cross-terms being zero on account of (11).
In the same way we obtain by means of (9)
u.= ul-?; 2
(14)
(15)
RELATIVISTIC MECHANICS
26
Ill,
Introducing
be written
and
(14), (15),
we
(16) into (10),
69
get an equation which can
(17)
Multiplying this vector equation by v yields, on account of
Hence, assuming that /
is
and
(13),
a monotonic function of the argument,
u l9
uv
(1 1)
(18)
This means, on the other hand, that the terms prov
in
to
(17) cancel each other and that the coefficients of i^
portional
and u l are equal. Consequently, equation (17) reduces to
i.e.
in (7).
[/(i)-/U/[f( -'
/f
Since, moreover, according to (7)
ut
the coefficient of
ux
/(u,)
If \ve are to
ut
l2
)+' l}(l-
and
in (19)
J(u 1
-u 1 )
0.
o,
-y.
must be
zero,
i.e.
(i- v *!c*WU\ul(l-v*lc*) + v*]}
have conservation of
(19)
(18),
2u x
D!
/c*)
momentum
in
any
(20)
collisions of the
kind considered, the function / must satisfy the equation (20) for
values of the independent variables u and
tional equation
way we
is
obtained by letting u in (20) tend to zero.
It is easily seen that the function f(u) given
In this
(2)
and
(21)
we thus obtain
m=
where we have put
(21) satisfies (20) for all
for the relativ istic
mass of a
___
V(i~^
/(())
mQ
(22)}
(
particle
m Q the so-called proper mass or rest mass of the particle,
with the mass assigned to the particle in Newtonian
The constant
identical
by
v.
with the velocity u
is
all
solution of this func-
get
values of u^ and
From
The
v.
mechanics, and the assumption preceding (10) obviously implies that
the rest mass is unchanged in the collision considered. For the momen-
tum
of the particle
we now
get,
according to
mo u
(1)
and
(22),
RELATIVISTIC MECHANICS
70
27
III,
work, kinetic energy
27. Force,
When
the velocity of the particje and therefore its momentum are
constant in time, this is taken as an indication that the particle is free.
If, however, the momentum of the particle changes, the particle is said
to be acted
upon by a force F which is equal
,/ n
T?
per unit time:
to the change of momentum
MT
-.
fc"4
A \
(24)
Equation (24), which for small particle velocities is identical with
Newton's second law of mechanics, should here be regarded as the
definition of a force in the relativistic region. It may be considered as
an equation of motion only when it has been stated how F depends on
the physical state of the system which is the cause of the change of
momentum
As
time
in
is
of the particle.
Newtonian mechanics, the work
defined by
where u
;=
done by the force per unit
(F.u),
(25)
the velocity of the particle. Further, the kinetic energy
the particle is defined by the equation
is
T of
rf
j-- A
at
---(F.u),
(26)
expressing that the change of the kinetic energy per unit time
the work A.
Using
(23)
and
d
(24) the right-hand side of equation (26)
mn u
ra n
u du\
equal to
may be written
mn u
is
du
dt)
or, since
u
u*"\
7J
.
'
c*(l-~u*/c
)i
dt
du
,
dt)
A =
/I
*^
,,.2/.,2U
J*
,7/1/1
^L
~.2ls,'2\\r
(27)
I
\
Introducing this expression into the right-hand side of (26), this
equation can at once be integrated and we obtain for the kinetic energy
of a particle with the velocity
u
<
where C is a constant of integration.
taken as zero for u
0, we get
C
and thus
~
m
--/7T
Since the kinetic energy
may
28 >
be
c2,
!!/^- m o^
(29)
If
RELATIV1STIC MECHANICS
27
Ill,
is
2
(u/c)
small compared with
and we then obtain,
c,
71
we can make an expansion
to a first approximation, the
in
terms of
Newtonian
expression for the kinetic energy
T = %m Q u 2
(30)
In the same way, all quantities of relativistic mechanics and the relations
between these quantities are identical with the corresponding quantities
and relations of Newtonian mechanics in the domain where the velocities
are small compared with
between
relativistic
It is interesting to note that all deviations
c.
and Newtonian mechanics are at
least of
second
why the early electron theory which was
based on Newtonian mechanics was able to explain all effects of first
order in u/c, which explains
When the velocity u approaches c, however, the deviations
between relativistic mechanics arid Newtonian mechanics are very large.
For u -> c both the mass (22) and the kinetic energy (29) become infinite,
showing that in mechanics also c plays the role of a limiting velocity.
order.
28.
Transformation equations for
momentum
energy and force
Consider again the two systems of inertia S and 8' corresponding to the
special Lorentz transformation (II. 24). The momentum and the
kinetic energy of a particle are then given in both systems
From
of the form (23) and (29).
by formulae
the transformation equations
(II.
45)
for particle velocities we then also obtain the transformation equations
for momentum and energy of a particle.
Now it is convenient to introduce a quantity E defined by the equation
=-=
T+
c*
-,-,-^^T
(1U*/C*)*
me*.
(31
differing from T by the constant amount
of the kinetic energy of the particle and frequently
The quantity E,
measure
c2
is
we speak
E of a free particle without, however, at the moment
A similar
physical meaning to the constant m c
simply of the energy
any
attributing
E = T
f
quantity
By means
(II.
48)
x
-\-m c
is
of (23), (31),
we then
get
u*
<J(l-u
introduced in the system $'.
and the relations inverse to (II. 45) and to
/c
--uc
Q
(l+vux /c z
u\+v
v
"-
Q (u'r
+v)
RELATIVISTIC MECHANICS
72
In the same
way we
obtain
A comparison of the equations
(II. 24')
(32)
with the Lorentz transformations
shows that the four quantities
Pv PS E\#
transformed in the same way as the space- time
Ps,
are
Thus we have,
coordinates x,
in analogy to the equations (15), (13),
and
(13') in
y, z,
t.
Chap-
/'2
*-%*=?
(23)
2
raj)C
(33)
ter II,
From
28
III,
and
(31)
Therefore
<
34 )
we see that the invariant (34) has the constant value
we have in any system of inertia
p*--=
c
E=
i.e.
-m\c\
2
c(ragc +j9
(35)
)i.
Hence, for the velocity of the particle
Since the quantities (33) also transform like the coordinates (x,y,z t)
by rotations of the Cartesian axes, the transformation equations for
9
momentum and
transformation
energy may, in the case of the more general Lorentz
be written in the vector lorm
(II. 25'),
p-p ^
E^
' 7/c
A/tl!_,-,<.-,
J
v>Lvv
Ir7'_l_
^M
(37)
!
immediately seen that (34), which is analogous to (II. 30), is a consequence of (37).
Let us now consider a systemef n free particles. If the momentum and
*>
(l)
l) 2
energy of the ith particle are denoted by p and E = T(t) -fw c
kinetic
and
w
T^
is
the
is
where
the
rest
mass
of
energy
respectively,
It is
momentum and
the ith particle, the total
fined
by
^
rr\
energy of the system are de-
2P< "'
2,
i
TO)
J-
>
ni ()
V
2,
i
<,
KKLATIVISTIC MECHANICS
28
Ill,
73
Since the transformation equations (32) or (37) are valid for^each
and as they are linear transformations, it is obvious
particle separately
that the same equations are valid also for the total momentum and
energy of the system. Thus we can take over the equations (32) and (37)
for a system of free particles, where p, E, and T denote the total momen-
tum and
energy, while
masses of the
w?
particles.
according to (38), is the sum of the proper
the transformation equations (32) and
From
immediately seen that, if the theorem of conservation of
momentum in a collision between the particles is valid in every system
of inertia, the total energy E must also be conserved in any system of
(37) it is
inertia.
particle, the equations (24)
Returning to a system consisting of a single
(26) may be written in the form
and
j = F,
(39a)
dtt
^=
(396)
(F.u).
Since the equations (39) are to hold in any system of inertia
deduce the transformation equations for the force F from the
we can
known
transformation properties of the quantities on the left-hand side of (39)
From the equations (II. 25') and (II. 26) we get
dt
_
~
Hi'
Thus, by means of
(v.u')/r
2
'
~J(i^*lc*)
(37),
^ dp dt'
dt'
dt
')[l^^
where F'
is
~~
'
the force in the system S'. In relativistic mechanics the conit has in Newtonian
cept of force has no longer any absolute meaning as
mechanics.
If
we
introduce the proper time T of the particle instead of
Minkowski force
and the
^
>
the equations (39) can, by means of
dp
dr
_,
p
M
I/
(II. 38),
dE
:il
,
dr
be written
.
u.
(42)
RKLATIVISTIC MECHANICS
74
III,
28
an invariant, the quantities {p,E/c 2} on the left-hand sides
of (42) transform like the space-time coordinates {x,}: the same must
be true, therefore, for the quantities {F/,/5 (F u .u)/c 2 }. This can also be
shown directly by an elementary calculation from (40), (II. 55), and
Since r
is
(II. 56).
Hyperbolic motion. Motion of an electrically charged para constant magnetic field
As mentioned before, the equation (39 a) can be regarded as equation
29.
ticle in
of motion only when it is known how the force F depends on the variables
of the physical system causing the change of momentum of the particle.
If the velocity of the particle is small relative to c, the relativistic equation must, however, be identical with Newton's second law and, in the
system 8 relative to which the particle has velocity zero at the
inertial
moment
considered,
we may assume that the
system $, we have v
in
u and
-*
is
identical with
calculate the force
the velocity of the particle relative to
By means of the transformation equations
F in an arbitrary inertial system S.
the Newtonian force.
we can then
force
S be
u'
u,
and we
(40)
Let
S' in (40) is the rest
then get for the force
if
(u
<
the Newtonian force. Decomposing the forces into two components respectively, parallel and perpendicular to u, thus putting
where
is
F
*
(43)
*
F,i4-F,
III* J_
-~
F?4-F^
-fJ. j_,
J.
||
can obviously be replaced by the simple equations
F L = F^l-* 2 /' 2
(43')
we
know
F
If
the Newtonian force
we can thus, by means of (43) or
(43'), calculate F in any system S. In this way we can show, for example
1
F,,=--F{|,
(Chap. V,
58),
that the force acting on an electrically charged particle
field E and a magnetic
travelling with the velocity u through an electric
field
is given by the Lorentz formula
F-
E + -(uxH)
(44)
the electric charge of the particle and u X
denotes the vector
product of the vectors u and H.
Now the left-hand side of the equation of motion (39 a) can, by means
of (1), (31), and (396), be written
where
dp
*
dt
e is
d(mu)
'
'_
dt
du
I
tyy\
dt
dm II
dt
~~
du
1
I
<yy\
dt
dE ||
dt
du
tyY*
dt
,
I
RELATIVISTIC MECHANICS
29
Ill,
where
(39 a)
we
75
the relativistic mass given by (22). Introducing this into
get the equations of motion in the form
is
m du
()
(45)
From
this we see that the acceleration of the particle in general has a
direction different from the direction of the force and, therefore, the
motion of the particle is more complicated in relativistic mechanics than
it is in Newtonian mechanics. Only if the force is constantly parallel
or constantly perpendicular to u, will the motion of the particle again
be relatively simple. We shall here treat an example of each of these
cases.
Let us
first
consider a particle which is acted upon by a constant force
has an initial velocity in the direction of the force.
F = m Q g and which
According to
(45) the particle will
then continue to move in the direction
of the force. Therefore the path of the particle will be a straight line and
this line as x-axis. (39 a) then reduces to
we can choose
u
If
we assume
first
integration
that the velocity
B
If
we
dx
is
dx
zero at the time
--
0, \\e
get
by a
at
, An
(46)
.
when
further assume that x
0,
we obtain by a second
integration
<47)
or
2
ff
we plot this motion in an ^/-diagram the equation (48) will represent
a hyperbola and, consequently, this motion is called a hyperbolic
motion. |
If
As long as (gt) 2 < c 2 we can neglect powers of (gt/c) higher than the
second, and we obtain from (47) the usual equation
,
:==
T?gt
representing the motion of a particle with constant acceleration.
t
M. Born, Ann.
d.
Phys. 30,
(1909); A. Sommorfeld, ibid. 33, 670 (1910).
RELATIVISTIC MECHANICS
76
For large
t,
i.e.
>
29
on the other hand, the increase in x
for large velocities,
slower than that according to Newtonian mechanics.
oo the velocity u given by (46) approaches the finite value c
with increasing
For
III,
is
independently of the value of g\ thus, even if a particle is attacked by a
very large constant force, it will never attain a velocity exceeding the
This
velocity of light.
An
p. 53.
electrically
agreement with the considerations
charged particle moving in a constant
is
in
in
21,
electric
field with a velocity parallel to the direction of the field represents a case
of the type just considered.
Let us now consider the motion of a charged particle in a constant
of the particle into comrespectively, thus putting
magnetic fidd H. Decomposing the velocity
and perpendicular
ponents parallel
U||-f-Uj_,
to
the force acting on the particle may, according to (44), be
written
F
i.e.
the force
and
(45)
is
= ^(uxH) =
c
(u
perpendicular both to u, u
xH),
L
(49)
and H. Therefore,
(F.u)
0,
assumes the same form as
in
Newtonian mechanics,
m-" = F =
at
i.e.
(Uj.xH).
(50)
denotes the relativistic mass which, however, in our case is
and u, according to (39 6), are constant
constant, since K and thus also
Here
in
in time.
From
of the
component of the acceleration
(50) it follows that the
direction of
H is zero,
field, viz. u\ \, is
{
in the
the component of the velocity in the direction
constant in time and, since also u
|u is constant,
i.e.
the same must be the case for the magnitude IUJL of the vector Uj
Therefore the path of the particle must be a helix having its axis in the
direction of the field
to
H will be
The projection of the path on a plane perpendicular
a circle with radius p determined by the condition that the
mu\jp in the circular motion must be equal to the force
centripetal force
(49).
Therefore,
we have
mu 2
=L
or
pL
mu<
Tr
-u\H
= -Hp.
(51)
Ill,
RELATIVISTIC MECHANICS
29
If the velocity of the particle
we
77
perpendicular to the direction of the
is
get simply
-Hp.
field
(52)
c/
This equation enables us to determine the momentum of a charged
and p. This method has been especially useful
particle by measuring
in cosmic ray analysis
30.
and in
j3-ray
and mass spectroscopy.
Equivalence of energy and mass
Let us again consider the system 2 X of n free particles. If
(p,
E) and
(p',E') denote the total momentum and energy of the system in the two
systems of inertia S and 8', respectively, the connexion between the
primed and unprimed variables is then given by (32) or (37). The invariant (34) will for such a system always have a negative value. For
w 2 c 2 and, keeping
n =. I this invariant is, according to (35), equal to
mind that (35) holds for each particle, it
2
2
p E 2 /c must be less than -]T m o >c f r n
in
is
>
immediately seen that
This means that
we
can always choose an
system 8' such that the total
inertial
p' of 2 t is zero in 8'. Viz., putting p'
relative velocity v of 8' and 8
in (37),
momentum
we obtain
v-^P
2
and, since p
E 2 /c <
2
0,
we have
E>
for the
(53)
cp so that the relative velocity v
determined by (53) is less than c, which must be required.
Such a system of inertia 8 in which the total momentum p
is
system of X t or the centre of gravity system since the
system Xj as a whole has the same mechanical properties as a particle
at rest relative to 8. Now let u be the velocity of 8 relative to an
called the rest
arbitrary system of inertia 8, then u also denotes the velocity with which
the system i^ as a whole moves relative to S.
If the s\ stem 8' in (37) is identical with the rest system 8, we get the
equations
^
-
^
~~
\/(T-i2/f~V
which express the
total
momentum and
particle
t
M=
energy of i^ as a function of the
It
of the system as the ratio
i]
J(T-u*lc*)'
now seems natural to define the total mass
between momentum and velocity in the same
velocity u of the system.
way as for a single
we must assign to
(cf.
a mass
equation
M given by
^1*
** = *~
)).
From
(54)
-JL
**
we then
see that
(55)'
v
RELATIVISTIC MECHANICS
78
This corresponds to a rest mass
can be written in the form
EO/C*
E/c
III,
which, by means of
= m +T/c
30
(38),
(56)
where m
denotes the
is the sum of the rest masses of all particles, and
total kinetic energy of the particles in the rest system.
With the expression (56) for the rest mass of Sj the equations (54) for
energy of 2 X become completely analogous to the
and
(31) holding for a single particle. From (56) we see that
equations (23)
the rest mass of S^ is larger than the sum ra of the rest masses of all
2
Thus the inner kinetic energy of the
particles by the amount T/c
system contributes to the inertial mass of the system by an amount
the
momentum and
2
equal to T/c
The invariant in (34) can, by means of (54) and
.
p2-E*/c*
(56),
be written
-Mlc*
(57)
in analogy to the equation (35) valid for a single particle.
(57) may also be used as a definition of the rest mass of D 1
Equation
The important conclusion that the inner kinetic energy of a system of
an inertial mass can now be extended to any
free particles corresponds to
kind of energy.
Starting from the assumed general validity of the conservation
theorems,
&m =
To prove
this
~.
c
important theorem consider a
(58)
'collision'
of the system
E! of free particles with another arbitrary physical system S 2 by which a
certain amount of energy and momentum is transferred from 2 X to S 2
..
Before and after this collision process the particles in Sj are free; therefore the total momentum and energy of S t before and after the process
are transformed according to equations of the form (32) and (37).
Subtracting
momentum
now
the transformation equations for the energy and
from the corresponding transformation
after the 'collision'
equations before the collision, we get if (Ap, AA ) and (Ap', AJ?') denote
the difference between the momentum and energy of EJ before and after
7
the process measured in
S and
S', respectively,
(59)
Ill,
RELATIVISTIC MECHANICS
30
79
If momentum and energy are to be conserved during the process
momentum and energy of the system 2 2 must be increased by
amounts Ap and A7 in S and by Ap' and AJ5' in S'.
In analogy to (34) it now follows from (59) that the quantity
is
an invariant.
When
the
the
this invariant has a negative value, it is again
8 in which Ap' ==
possible to find a system of inertia S'
and if u is the velocity of S relative to S, we get from (59)
Ap
0,
AK
u A//r 2
A comparison of these equations with (-2) and (23) shows that we must
assign to the energy &E transferred to H a rest mass
2
(02)
C"
and a mass
Am
relative to 8,
which
is
change of momentum of 2 2 during the process is the same as if a
material particle with the rest mass Am and the velocity u relative to
for the
S had been added to the system X 2 The invariant (60) can now be written
.
|Ap|2_^|l
in analogy to (35).
mass
^ _(Am
2
)
(64)
This equation gives a simple expression for the rest
of the transferred energy.
H 2 was a general physical system so that the transferred energy
have
any form, we see that formula (63) must hold for any type of
may
energy. The system 2 2 may, for instance, be an electromagnetic field such
that the energy transferred from Sj has taken the form of electromagnetic
radiation. Thus the transformation equations for the energy and momenSince
tum
Do
is
of electromagnetic radiation must be given
a body which transforms the absorbed energy
also that the heat energy of a
body contributes
by
(59).
Further,
if
AA into heat, we see
1
to its mass, so that the
increased by heating. If, finally, 2 2 were a system
converting the transferred energy into potential energy, it follows that
inertial mass has to be assigned also to the potential energy of a system.
mass of a body
From
is
the preceding argument, in particular from equation
mass of a certain amount of energy
follows that the notion of
(64), it
E has a
RELATIVISTIC MECHANICS
80
30
III,
meaning only if we also know the momentum Ap connected
with the energy, and for this mass to be real the left-hand side of (64)
must be negative. Only for
well-defined
>c|Ap|
|A#|
may we speak
(65)
of a rest system and, consequently, of a certain
velocity and of a real mass of the energy.
In the discussion above E may be negative, so that we actually have
at
all
to deal with a transfer of energy to the system S^ Thus, considering a
process in which the total momentum p 2 and energy 2 of 2 2 are trans-
X we see that the transformation equations (59) must hold for
the total momentum and energy of an arbitrary system. If, furthermore,
ferred to
the relation
p*-~<
holds, the system has a real rest
(}
mass defined by
(64), i.e.
by
p|_.
C"
In the case where p\'K\l< ^
will also be zero.
t
>
H2
(67)
equal to zero the rest mass of the system
is
where the density of field energy is a homomust always
be fulfilled. If E2 were less than cp 2 we could find a system of inertia 8'
in which E'2 =and p 2 ~/~ 0. For this purpose it would be sufficient to
v
in
the
transformation
give
equations (59) for p 2 and E2 the value
If
is
a system of
fields
geneous positive definite function of the field variables, (66)
,
v-A'
P 8 /pI
(|v|<c).
According to the relations inverse to (59)
(08)
we should then have
According to the assumptions made above regarding the dependence
of the energy density on the field variables, E'z can, however, be zero only
if the field itself vanishes and in this case the momentum of the field
p'2
must
also be zero. It
is
therefore natural to assume that the relation
(66) holds for any macroscopic physical system. As a simple example
of a system where the sign of equality holds in (66) we have the case of a
train of electromagnetic plane waves. According to (67), such a system
c 2p 2 /E2
must therefore have the rest mass zero and a velocity u
c
in
any system of inertia.
Now it may also be shown that, conversely, any material particle with
UKLATIVISTIC MECHANICS
30
Ill,
m must represent an energy E = me
system of the particle an energy E = m
the mass
rest
81
and
in particular in the
c2
This statement ob-
viously has a real meaning only when a process exists in which the
energy represented by the mass of the particle can be transformed into
another form of energy such as the kinetic energy of other particles.
We cannot know in advance whether such an 'annihilation process'
actually does exist in nature, but we can show that if it exists under
certain conditions and if we require that the principle of relativity as
well as the theorems of conservation of momentum and energy shall
hold for this process, the amount of energy liberated by annihilation of
the mass m will be equal to E
m c2
To prove
this statement,
we assume that the momentum and energy
are transferred to the previously
of
free
1
particles. Consider again two systems of
inertia 8 and /S, where 8 is the rest system of the particle. If (Ap, Ai7)
and (Ap, A/) denote the transferred momentum and energy measured
by the annihilation process
liberated
considered system
in
S and $,
respectively,
we have, according
Ap = Apo-, U~
where u
^J(1U
to (59)
/C")
the velocity of the particle relative to 8.
momentum of the particle is zero in the rest system and
since, moreover, momentum and energy of the particle are zero in any
system of inertia after its annihilation, we obviously have
is
Since the
Ap
where
EQ
denotes the
A# = E
0,
unknown energy content
(70)
of the particle at rest
before the annihilation. Furthermore, Ap is equal to the
of the particle relative to 8 before the annihilation
momentum p
Substituting from (70) and (71) in (69)
ra
The
c2
u/c
(72)
total energy content of a free particle with the velocity
then be
3595.60
we obtain
__
EQ = m
or
E = EQ +T =
C2
2
me 2
u must
(73)
RELATIVISTIC MECHANICS
82
where we have made use of
(29)
The quantity
meaning. The energy
and
in (31) has thus a deeper physical
III,
(22).
30
introduced
(72) is called
the proper energy or rest energy of the particle.
We have thus obtained a general proof of Einstein's famous formulaf
E = me
(74)
which states that any energy E has an inertia corresponding to a
mass E/c and that any mass m represents an energy me This theorem
2
of the equivalence of energy and mass is one of the most important
results of the special theory of relativity. It should be noted that the
mass involved
in this
theorem
is
the inertial mass.
One of the
basic
assumptions of the general theory of relativity is, however, (cf. Chap.
VIII, 83), the equality of inertial and gravitational mass, so that we
shall
have to ascribe to an energy of amount
given by
E also a gravitational mass
(74).
Mass
31. Inelastic collisions.
of a closed
system
of particles
Before proceeding to a discussion of the experimental verification of
relativistic mechanics we shall, in the present section, consider a few
simple examples illustrating the general theorem of the equivalence of
mass and energy
Let us
two clay
first
consider a completely inelastic head-on collision between
same rest mass Q which in the inertial system
balls of the
travel along the same line with velocities of equal magnitude but
opposite in direction. The total momentum and energy in the centre of
S'
gravity system
before the collision are then
p
where
which
T
iS'
0}
//;o
2m
c2
+T,
(75)
the total kinetic energy. In a system of inertia S relative to
has the velocity u, we thus have, according to (37) and (75),
is
for the total
momentum and
energy before the collision
2
(2m +2*>/c )u
By
ball
the inelastic collision, the two clay balls will unite and form one large
on account of the theorem of conservation of momentum applied
;
in /S, this ball will
t
Kmstoin,
JRelativitatspnnzip
(1911).
have zero
Ann d
3
momentum and thus also zero velocity in 8.
Phys. 18, 639 (1905), 23, 317 (1907), H. A. Lorentz, Das
Haarlemer Vortrage, Leipzig 1914, id, Amal. Versl. 20, 87
RKLATIVJLST1C MECHANICS
31
Ill,
83
The original kinetic energy T
of the particles in S is converted into heat
of energy, the amount of heat
conservation
of
the
on
account
and,
developed
is
The day
where
tion of
and
ball after the collision
is
the velocity u relative to S,
_.
the rest mass of the ball after the collision. As the conserva-
momentum theorem must
(76 a)
/77\
710
now has
momentum
i.e.
must be
hold also in S, the expressions (78)
equal. Thus, applying
(77),
we
get
Jlf
-----L>
(79)
-|
\j
e.
the rest mass of the ball
{}
equal to the sum of
the
mass of the heat
by
after the collision
the rest masses of the original balls increased
is
energy.
For the energy of the ball after the collision in tf,
to (79) and (77),
C*
- ~" C T
-
which
~'
we obtain, according
'
<>
equal to the total energy (7()/>) of the balls before the collision,
in accordance with the theorem of conservation of energy in #. For the
is
difference between the total kinetic energy before
we
and
after the collision
obtain, however,
I
i\
no
(81)
by means of
(7(5/>), (79),
and
(77).
This difference
is
thus an invariant
independent of the system of inertia in which the kinetic energies are
calculated.
To
give an illustration of the inertia of potential energy we now conX consisting of a certain number of particles which are
sider a system
held together by attractive forces.
Let us assume that there exist
systems #' in which all particle velocities are small compared
with the velocity of light such that in *V \\e can use the non-relativistic
Newtonian mechanics as a good approximation. Neglecting the typically
inertial
atomic phenomena connected with the existence of Planck's quantum of
action, we may consider an atomic nucleus to form such a mechanical
system, the atomic nuclei being built up of elementary particles, the
nucleons, which are so heavy that their velocities can generally be
RELATIVISTIC MECHANICS
84
31
III,
regarded as small compared with c. This assumption means that the
proper times of the separate particles in 2 are practically identical and
equal to the time /' in S' and, furthermore, that the forces between the
particles can be considered to a first approximation to be functions of
If, moreover, the forces are conservative
can
in
S'
be
expressed as gradients of a potential function
they
which is a function of the position coordinates of the particles. According
the positions of the particles.
forces,
system will now move in
such a way that the sum of the kinetic energies and the potential energy
to
is
Newtonian mechanics the
constant in time,
particles in the
i.e.
T'
+V
=--
//'
constant.
(82)
However, the total kinetic energy is, of course, not constant in general.
so
We now adjust the arbitrary constant in the potential energy
that
--
when the particles
are so far from each other that the forces
Thus V is obviously ^negative
are bound to each other.
are zero.
in
any
state
where the
particles
Among the possible systems of inertia S' we now choose that system
S in which the centre of gravity of X is at rest. The sum p of the
momenta of the particles is then obviously constant and equal to zero
in H. In a system of inertia 8 relative to which /S is moving with the
velocity u in the direction of the #-axis, the sum p of the momenta of the
particles
is,
where again
according to
w?
is
(32),
given by
the sum of the rest masses of all the particles. Since the
proper times of the particles of the system are practically identical, p
can be regarded as a function of a well-defined time variable. In contrast
to the case of a system of free particles, p is not constant in time since
the kinetic energy T occurring in the expression (83) for p is time-
dependent.
However, p
is
not the total
momentum of S in
into account the fact that the potential energy
The total momentum P in S is therefore
where
T-\- V
is
we have to take
rest
mass V/c 2
the total energy in the rest system. In contrast
vector P is obviously constant in time,
a
for
closed
required
system with no external forces. If the
7/
to p, the total
as
S, since
V has a
is
momentum
RELATIVISTIC MECHANICS
31
Ill,
85
away from each other that they can be regarded as
we must have P - p, a condition which determines uniquely the
particles are so far
free,
constant in the potential energy.
From (84) we see that an atomic nucleus which moves as a whole with
the velocity u relative to a system of inertia
"=
with a rest mass
total
momentum
;,->
^ m +H/c
Q
(86)
7/ is the binding energy
negative and A A ^the amount of energy which must be transferred to
For stable nuclei
//
is
of the nucleus, i.e.
the nucleus in order to disintegrate
WQ
particles. For the quantity Am
we
8 has a
it
completely into its constituent
3/ the mass defect of the nucleus,
,
ha\e, according to (86),
A
Am = ^.
(87)
c2
This fundamental relation connecting the binding energy and the mass
defect, which is a special case of Einstein's formula (74), has now been
verified experimentally with great accuracy in
many
nuclear reactions
32).t
(cf.
Experimental verification of relativistic mechanics
26 we saw that, if the theorem of conservation of momentum is to
hold in any elastic collision between two particles, the variation of the
mass with velocity must be given by (22). The next question is whether
something like conservation of momentum and energy exists at all for
32.
In
large velocities. This question can only be settled by experiments.
Direct experiments of this kind were performed by Champion, J who
investigated the collisions between rapidly moving electrons (j3-particles)
and
elecl
rons at rest in a Wilson chamber. Let us consider more closely
1 with velocity U 1 and momentum
2 with velocity zero, i.e. with
an
electron
and
elative to a systom of coordinates S in which the
such a collision between an electron
pl
the
raQU^l
ftf/c )-*
momentum p 2
cloud chamber
is
at rest. In the collision a certain
momentum
is
trans-
p l and p 2 be the momenta of the two particles
and the angles between the direction of the
incident electron and the vectors p l and p 2 respectively (cf. Fig. 11).
ferred to particle 2. Let
after the collision, and 6
<f>
t The possibility of such an effect was first discussed by P. Langevin, J. de Phys. (5),
3, 553 (1913).
J F. C. Champion, Proc. Roy. Soc. A, 136, 630 (1932).
RKLATIVTSTIC MECHANICS
86
32
III,
Application of the conservation laws of relativistic mechanics then leads
to a simple relation between 6 and <, which we shall now deduce.
It is convenient here to make use of the fact that the theorems of
conservation of
(if
they hold at
momentum and energy are
valid in
any system of inertia
For convenience we choose the Cartesian axes
all).
in
KKJ. 11
the laboratory system 8 such that p t is parallel to the #-axis and p t lies
in the #y-plane. Then, if the momentum is conserved, p 2 also must lie
We now introduce the
momentum p'
pi+Pa
in the a:/y-plane.
which the total
centre of gravity system S' in
8' is moving relative to 8
-= 0.
with a certain velocity v in the direction of the .r-axis.
From the reciprocal
equation to (32 a)
which holds
for
each particle separately as well as for the total momenE of the system, we then get, since p -- p l and
tum p and energy
E = E^m^c 2
c
+m
2
(l~^f/r )"s the following expression for v:
Pi
Since p'2 -pi, the two particles have the same initial velocity u'
relative to 8' Since, furthermore, particle 2 was at rest in S, the velocity
'.
u' relative to 8'
must be equal
v,
IE'
to the relative velocity r of 8'
and
8,
i.e.
(89)
Applying now the theorems of conservation of momentum and energy
we obtain for the momenta and energies
in the centre of gravity system,
Ill,
RELATIVISTIC MECHANICS
32
87
of the particles after the collision
I>2
=
=--
Pa
Pi
Pi
E'i
E*
\E'
..g/-^
-/,-,
._
Consequently the two particles
will also
have the same velocity v relative
we have
to S' after the collision and, according to (90),
-*?=*.
From
Fig.
we
<..,
see at once that
4~
tan 9 ~-
tan
<f>
2"
Plx
Using the transformation equation
particles after the collision,
we
P2jr
(32) for the
momenta
of the
two
obtain by means of (90) and (91)
Introducing the expression (88) for
?;
tan0tan<)&
where we have put
-y l
into (92)
we
finally get
"
(93)
71+
(lul/c
(94)
)-*.
Thus, the product of tan0 and tan^ is independent of the rest mass of
the particles (as long as they are equal) and is a function of the velocity
of the incident particle alone.
In the limit -> oc, we obtain the corresponding formula of Newtonian
mechanics. In this limit,
c,
yl
and
tan#tan</>
1,
.e.
or
6+<f>
ITT.
(95)
According to Newtonian mechanics the directions of motion of the
two particles after the collision should thus be perpendicular to each
other.
This well-known effect can be observed in collisions between
and has also been verified in collisions between
and helium nuclei in a Wilson chamber, the velocity of the
billiard balls, for instance,
a-particles
RELATIVISTIC MECHANICS
88
a-particles being so small
compared with
may be applied in this case.
approaches the velocity of
When
light,
HI,
32
that Newtonian mechanics
the velocity of the incident particle
however, y l
tan 6 tan
cf>
<
>
and, thus,
1,
which shows that the angle between the directions of motion of the
particles after the collision in this case will be smaller than \TT.
is well suited to an experimental test
Cloud-chamber pictures of the collision of
j8-particles with electrons at rest in the chamber allow a direct determina-
This characteristic phenomenon
of relativistic mechanics.
tion of the angles 9 and <. It turns out that in many cases #+</> is considerably smaller than 90. If, in addition, the velocities, and consequentlv
also
y 1? for the incident /?- particles are known, we arrive at a direct test of
the relativistic formula (93). Measurements of that kind were performed
by Champion, who found agreement with formula (93). This may be
*)*
regarded as an experimental proof of the theorems of conservation of
momentum and energy in collisions between electrons, and thus also
of the relativistic formula (22) for the variation of mass with velocity.
The accuracy of these measurements was quite sufficient to disprove
the formula for the variation of mass with velocity put forward by
Abraham} before the development of the theory of relativity.
The equations of motion (39^) and (44) can also be tested directly by
measuring the deflexions of rapidly moving electrons in electric and
magnetic fields. In the case of a constant magnetic field perpendicular
to the direction of motion of the electron, the previously derived equation (52) should hold, i.e.
mU =
(96)
e
If,
H?.
on the other hand, the electron traverses a constant
electric field
perpendicular to the initial velocity u of the electron, the electron will,
according to (45) and (44), have a constant acceleration g
eE/m in
the direction of the field as long as the deflexion from the original path
is
In the time
small.
distance
l/u
which the electron requires to traverse a
in the field, the electron
is
thus deflected a distance
-*-*
perpendicular to the initial direction of motion.
If, therefore, the electron traverses both a magnetic and an electric field
of the above-mentioned kind, simultaneous measurements of
f See ref., p. 85.
M. Abraham, Ann.
d. Plnjs.
//, p,
E,
10, 105 (1903).
I,
Ill,
RELATIVISTIC MECHANICS
32
89
and AZ allow us to determine the variation of m with u by means of (96)
and (97). Such experiments were performed by a large number of investigatorsf and the results obtained in the later experiments are in close
agreement with the relativistic formula (22). Simultaneously the
experiments may be regarded as a test of Lorentz's expression (44) for
the force acting on an electrically charged moving particle.
If we know the charge of the particle from other experiments, measure-
ments of the deflexion of the
particle in
known
electric
and magnetic
allow us to determine the absolute value of the mass of the particle.
This is used in the so-called mass spectrograph, which permits a .very
fields
mass of atomic nuclei.
While thus the relativistic equations of motion of fast charged particles
were experimentally proved at an early time, it was not until recent years
possible to test the equations (74) and (87) which express the equivalence
of energy and mass. This is understandable if we bear in mind that the
change in mass of a body due to its potential energy or due to heating
precise determination of the
according to these equations in general will be negligible compared with
the total mass of the body. The situation is different, however, when
Einstein's relation
applied to single atomic nuclei.
is
From the results obtained at the end of the preceding section we should
expect the mass of an atomic nucleus in its ground state to be always
smaller than the sum of the masses of the constituent nucleons. According to (87) this mass defect should be particularly large for the stablest
nuclei with large binding energy. Measurements of nuclear masses by
means of the mass spectrograph have confirmed this result and in some
cases the
mass defect amounts
to several per cent, of the
mass of the
whole nucleus.
As soon
as one succeeded in producing nuclear reactions in
which
nuclei with a given mass defect are transformed into nuclei with other
mass defects it also became possible to test Einstein's formula (74)
experimentally. Here we shall only discuss one single example of such a
process. When lithium is bombarded by fast protons (Oockcroft and
Walton, 1932J) it happens that a proton (}H) penetrates into, a lithium
nucleus
(JLi),
thus forming a
quickly splits into
two
compound nucleus which
fast ex-particles (|He).
is
unstable and
This process can be
the numerous paper** wo quote W. Kaufmann, Gott Nadir, niath.-nat.
H. Bueheioi, Veth. d. Dculxchen Phy*. Ges. 6, 688 (1908),
G. Neumann, Ann. d. Phys. 45, 529 (1914); Ch. K. Guye and Ch. Lavanehy, Arch, de
Gcrlach
Geneve, 41, 286, 353, 441 (1916). See also the comprehensive account by
in Hdb. d. Phys 22, 61 (1926).
J J. D. Cockeroft and G. T. S. Walton, Pioc. Roy. floe. A, 137, 2J9 (1932).
t
Among
Klasse, p. 143 (1901); A.
RELATIVISTIC MECHANICS
90
32
III,
expressed by the equation
IU+\H - JHe+JHe.
(98)
of the nuclei entering in this process are known very well
from mass-spectrograph measurements. According to the latest measure-
The masses
ments the mass of Li
to 16.
Now,
if
we
is 7
66 if the mass of the oxygen atom
is
put equal
also content ourselves with four decimal places for
the hydrogen nucleus and the helium nucleus, the corresponding figures
for JH, |He are 1-0076 and 4-0028, respectively, f Thus the loss in mass
during the process
m=
mass
units, or in
is
2x4-0028
7-0166+1-0076
0-0186
grammes,
m=
10- 25 gin.
0-309
me*
27-7
(99)
mass represents an energy of
According to Einstein's relation this
0- erg,
(100)
which after the process must appear as kinetic energy of the
a-particles
By means of the range-energy relation the kinetic energy of the particles
now be determined from measurements of the ranges
particles. Newer very precise measurements^ give for the
can
between the total kinetic energy of the
a-particles
of the
ex-
difference
and the kinetic energy
of the incident proton the value of
!7-28()-03MeV
The agreement between
(100)
---
and
6
(27-60-()5)x 10~
(101)
is
excellent.
erg.
(101)
The deviation
is
than the possible error in the accuracy of the measurements, the
error in the determination of the mass of the lithium nucleus involving
less
an error in me 2 of
x 10- 6
Einstein's equation (74) can thus be
regarded as being verified experimentally with high accuracy, the error
0-2
being less than 1 per cent.
While the mass corresponding to the heat developed during a usual
chemical process is immeasurably small, the amount of heat developed
in a nuclear reaction pile
is
so large that the corresponding
of the order of
mass may be
the fission of a uranium nucleus,
many grammes.
where the nucleus is divided into two fragments, an energy A A is
2
developed whose mass Am = A//c amounts to about a thousandth of
the mass of the uranium nucleus. By complete fission of all the nuclei
in 1 ton of uranium the mass of the heat developed would thus be of the
Tri
order of magnitude of
kg.
K. T Bambndge arid E. B. Jordan, Phy.s. Rev. 51
M. S. Livingston, Rev Mod. Phys 9, 370 (1937).
J N M. Smith, Jr., 7%*. Rev 56, 548 (1939).
f
,384 (1937)
.see
aho H. Bethe and
KELATIVISTTC MECHANICS
32
Ill,
91
Modern atomic
physics, on the other hand, could also verify the
equation (72) according to which a particle of mass w represents an
EQ
c2
This statement can be tested only (and therefore has
a real meaning only) if a process exists in which the particle is annihilated
completely. After the discovery of the positive electrons, the positons,
energy
in 1932f
it
became
do exist in
and
a
negaton) are
positon
clear that such annihilation processes
which a positive and a negative electron
(a
annihilated, in accordance with Dirac's theory of electrons. J Since both
particles have the same mass m^ the amount of energy liberated in such a
2
This energy is
process should, accoiding to (72), be equal to 2w r
emitted in the form of electromagnetic radiation and by measurements of
.
the energy of this radiation equation (72) could be verified. The reverse
process, in which electromagnetic radiation (light quanta) is transformed
into pairs of positive
light
electrons, is also possible provided the
participating in each individual process
quanta
energy of
and negative
2r/?
have the necessary
2
.
Summarizing, we can state that all consequences of relativistic mechanics have proved to be in complete agreement with the measurements.
The objection which could be raised against this statement is that all
these measurements were performed on atomic particles, electrons and
nuclei, etc., which in view of the existence of the quantum of action
cannot be treated by means of
classical
mechanics. Since, on the other
only a special case of quantum mechanics,
the experiments mentioned above simply show that such consequences
as the theorem of the equivalence of energy and mass must be valid
hand, classical mechanics
beyond the domain of
is
mechanics. In the general proof of this
30, the system H 2 was in fact a quite general physical
classical
theorem, given in
system without any limiting assumptions regarding its constitution.
Moreover, in all experiments on the deflexion of electrons in a macro-
scopic electromagnetic field we are concerned with a case where quantum
mechanics goes over into classical mechanics so that these experiments
can be regarded as a direct verification of the fundamental equations of
classical relativistic point mechanics.
Andeison, faience, 76, 238 (11)32) /'////*. Her. 43, 4!H
9
P. S. Oochialmi, I tor Rot/. Noc A, 139, 699 (1933)
f C. J>.
and G
933)
P.
M. S Blockrtt
M. Dirac, The Principle* of Quantum Mechanic*, 3id ed., (Kfoid 1947, 73.
S. H. Noddei moyoi
See also
Pht/tt lt<v. 43, 1034 (1933).
Kasetti, L. Meitner, and K Philipp, Natunv. 21, 286 (1933), J. Cune and F. Johot,
P. A.
C D. Anderson and
K.
CM.
196, 1581 (1933).
IV
FOUR-DIMENSIONAL FORMULATION OF THE
THEORY OF RELATIVITY: TENSOR CALCULUS
Four -dimensional representation of the Lorentz transformation
IN a definite system of inertia S an arbitrary event is characterized by the
33.
four space-time coordinates (x,y,z,t). In another system of inertia S'
the same event is characterized by four other numbers (#',?/', 2', '). If
we assume that the origins of the ( 'artesian coordinates in the two systems
S and
8' coincide at the time
t'
0,
the connexion between these
space-time coordinates is given by a homogeneous Lorentz transformation, i.e. by a homogeneous linear transformation leaving the quantity
s 2 (II. 13) invariant, i.e.
s2
in ^
x 2 +y 2 +z 2
xl
x'i
x,
x',
c 2t 2
x2
y,
x'2
= y'
x' 2 +y' 2 +z' 2
a?
x'3
~
=
2,
#4
2',
x\
=
=
c 2t 2
s' 2
(1)
ict
ict',
where i
1) is the imaginary unit, the homogeneous Lorentz trans<j(
formation can thus be characterized as a homogeneous linear transformation
(=1,2,3,4),
(3)
k- 1
which
satisfies
the condition
=!>;= 2 x?.
i
Here the
(4)
oi
ik depend only on the angles between the spatial
and on the relative velocity of the two systems of inertia.
Since the coordinates x 1% x 2 x 3 are real, and # 4 is purely imaginary in
axes in
coefficients
S and
S'
every system of coordinates,
oc
while
if
iK
and a 44 are
a l4 and
4/c
real,
are purely imaginary,
and K denote arbitrary index values between 1 and 3.
and (x\) were real numbers, they could be
If the variables (x t )
inter-
preted as Cartesian coordinates of a point in a four-dimensional Euclidean space. Then the transformation (3) would represent a simple rotation
of the Cartesian system of coordinates in this space, the distance
(4)
THE THEORY OF RELATIVITY
33
IV,
from the observed point (#J to the origin
93
(0, 0, 0, 0)
being invariant
by such a rotation.
Now
natural, even if # 4
it is
and
x\ are not real, to introduce a four-
dimensional space whose points are defined by the coordinates (xj.
Since any event in physical space is characterized by a definite set of
numbers (# t ) in every system of space-time coordinates, each event is
thus depicted in a definite point of this abstract four-dimensional space.
This space, which was first introduced by Poincaref and Minkowski,J
continuum or simply the four-space or (3+1)that
the four dimensions of the space are not comspace, hereby recalling
A
pletely equivalent.
homogeneous Lorentz transformation (3) can
is
called the space-time
thus be interpreted as a rotation of the system of coordinates in the
(3+ l)-space. The invariant form (4) is naturally called the square of the
four-dimensional distance between the event point (x t ) and the origin
(0, 0, 0, 0). In view of the formal similarity to a Euclidean space, all
usual geometrical notions can
geometry
now be used
in the
in this space is called pseudo-Euclidean.
(3+1) -space. The
The deviation from
Euclidean geometry is characterized by the circumstance that the
distance (4) can be zero without all (x t ) being zero. All points whose
distance from the origin is zero form a surface described by the equation
6-2
= ^ x\
=-
2
2
x*+y +z 2 -c
0.
(6)
This surface is called the light cone, since the equation (6) describes the
propagation of a spherical light wave starting from the origin
x
z
y
at the time
0.
The
light
cone divides the (3+l)-space into two
by the inequalities
invariant separate domains a and b characterized
s2
s2
and
x 2 +y 2 +z 2 -cW
x 2 +y 2 +z 2 -c 2 t 2
<
>
(la)
(76)
0,
respectively. For events in the latter domain we can always by a Lorentz
transformation introduce a system of space-time coordinates S' in which
t'
0, i.e.
the event
(a?J
(x,y,z,t) is simultaneous
(0,0,0,0) in the system S'. Such a transformation
two events in the domain (la).
is
with the event
not possible for
An arbitrary motion of a material particle can be described
tions of the form
t
x
i, 2, 3),
(
^=
where the X
t
{
(X^)
by equa-
are definite functions of the time variable x 4
H. Pomcare, Rend. Pal 21, 129 (1906).
H. Mmkowski, 'Raum und Zeit', Phya. ZS. 10, 104
(1909).
(8)
.
These
FOUR-DIMKNSIONAL FORMULATION OF
94
33
IV,
equations represent a curve in the (3-|-l)-space which we shall call the
time track of the particle. In the case of a uniform motion the functions
x^x^) are linear and the time track is a straight line.
the origin, it must evidently Jie entirely inside the
If it
goes through
domain (la)
since
the velocity of the particle is always smaller than the velocity of light.
Then we can always introduce a system of coordinates $' such that the
#4-axis coincides with the time track of the particle. In ordinary physical
space (3-space) this means simply that we can always introduce a system
of inertia which follows the particle in its motion.
Let us now consider two arbitrary events with coordinates (xj and
(:r ) in *S
By transition to another system of coordinates 8' both sets of
Y
coordinates are transformed in the same way, viz. by equations of the
This means that the differences
(3) with the same coefficients r\ lA
form
(x l
between these coordinates are also transformed according to
these equations, so that
I (a,--?,)
I (*'-%)*
(9)
must be an invariant. The quantity (9) represents the square of the fourdimensional distance between the event points (x and (I*,). Since both
2 #? and 2 ** arc invariants it follows at once from (9) that
t
also
is
an invariant.
Using the expressions
(3) for x\
and
in the right-hand side of (10)
x\
we obtain
2 (2^
t
According to
:r
'
'
(10) this expression
dent values of the variables
when the
^ 2 (l>,/^mH^ml,m
a
/)(2
m *m^m)
(xt )
'
must be equal to
and
This
(x m ).
is
J xi^i f
a " indepen-
possible, however, only
coefficients satisfy the relations
2 a //^m
8 /m
(M)
uhere
is
8 /wl -=
(0 fur
]
lor
i~
m
m
,,
rt
(12)
the well-known Kronecker symbol.
The relations inverse to (3) are simply
for
if
we use
(3) in
2 x \<*ik
i
=-
the right-hand side of (13)
2
1,1
<*ti
xi<*tk
we
get
= 2 */(2 a 7/fA-) = 2 x
i
<.
by means of
8M
x k>
(11)
IV,
THE THEORY OF RELATIVITY
33
Using
(13) in the left-hand side of (10)
argument
as in the derivation of
The conditions
(11)
and
1 1
95
we then obtain by the same
(14) for the coefficients a ik the so-called ortho,
merely express the fact that the homogeneous
gonality
Lorentz transformations represent 'rotations' of the system of corelations,
ordinates in (3+1) -space.
For the determinant formed by the scheme of coefficients
a, A., i.e.
(15)
we
obtain, using (14)
and the multiplication
rule for determinants,
.e.
(15')
For proper rotations m must be equal to
1, since a rotation may be
in
a
continuous
and
the
identical
transformation x\
x%
performed
way,
has a scheme of coefficients whose determinant is 1. However, for
reflections where one or three ol the axes change their sign, a will have
the value
The
coefficients (a iA> ) for the special
are given
Lorentz transformation
(II.
24)
by the following scheme
y
1
(16)
\ivy/c
where y
(l
v /c )-*. It is easily verified that (16) satisfies the ortho-
gonality relations (11) and (14) and that a -- +1.
The special Lorentz transformation can also be written in the form
(17)
where we have put
COSi/r
y,
iiy/c,
tant/f
ivfc.
(18)
FOUR-DIMENSIONAL FORMULATION OF
96
Formally the equations
the 'angle of rotation'
(17) represent
i/r
is
a rotation in the
(^ 1
IV,
33
^ 4 )-plane, but
a purely imaginary quantity.
Lorentz contraction and retardation of moving clocks in
four-dimensional representation
Any event which takes place on the r-axis in the system of inertia 8
34.
is
represented by a point in the (.r^) -plane in (3+l)-space.
S'\
Fi
which
is
Fig. 12,
drawn
as
if
12.
the time variables # 4 # 4 an d the angle
,
>
ifj
were
real,
gives an illustration of the Lorentz contraction and the retardation of
clocks. The lines L l and //.,, which are parallel to the x 4 -axis, represent
the time tracks of the end-points of a measuring-rod at rest
xAl
ar'-axis, the rest length being / -~ xAt
on!
the
of the measuring-rod relative to S will then be equal to
the difference in the ^-coordinates of two event points A 2 and A* which
The length
are simultaneous (and thus have the
in S,
i.e.
formal relations from Euclidean geometry are valid here also,
follows from the figure that
Since
it
same # 4 -value)
all
where we have used
---
(1H).
sec^
Equation
(l-?;
/c
(19)
-,
(19) is the
Lorentz contraction
formula. Similarly, if we consider a measuring-rod at rest on the #-axis
of the system /S, the time tracks of the end-points are given by the lines
and 2 parallel to the .r 4 -axis. A consideration of the triangle B l B 2 B%
l
thus leads again to the equation (19).
In the same way, we may get a geometrical illustration of the clock
retardation effect. ( 'onsider a clock at rest in the system S' its time track
;
IV,
'
THK THEORY OF RELATIVITY
34
97
given by a straight line parallel to the x^-axis, and the proper time r
be this line, and let
a measure of the length of this line. Now let
Cl C2 be that part of the line which corresponds to the proper time r.
is
is
The corresponding time t in S is obtained as the projection of C^C^ on
the # 4 -axis. Then a consideration of the triangle C^C^C^ gives immedi-
(20)
ry,
in accordance with formula (II. 36) for the clock retardation.
On the other hand, if we have a clock which is at rest in the system $,
a consideration of the triangle D1 D2 D^ gives again formula (20) for the
connexion between the time /' of the system 8' and the proper time of the
clock.
In this representation the Lorentz contraction and the clock retardation effect appear as a kind of perspective shortening of measuring-rods
and time
sentation
intervals.
is
However,
it
quite formal; this
'angle of projection'
ifi
is
must be emphasized that such a reprealso manifest from the fact that the
is
an imaginary quantity.
of the
On
the whole, the
four-dimensional
epistemological significance
representation
should not be exaggerated. In spite of the formal symmetry between the
description of space and time in the theory of relativity there is still a
fundamental physical difference between the space and time variables.
This difference is intimately connected with the difference of the
measuring instruments, clocks and measuring-rods, which are required
for the physical definition of these variables. Therefore
it is
not possible
by any admissible 'rotation' (3) satisfying the conditions (5) to transform the time axis into a space axis Only those rotations have a physical
meaning in which the r 4 -axis remains within the domain (7 a), i.e. inside
the light cone
(6).
Covariance of the laws of nature in four-dimensional
formulation
35.
In spite of its purely formal character, the four-dimensional representation has been of great significance in the development of the theory
of relativity, since it allows us to express the covariance of the laws of
nature under Lorentz transformations in a particularly simple way.
Each law of nature expresses a certain relation between physical quantities. These quantities are defined
by the procedure to be applied in their
measurement. Let A, #,... be such a set of physical quantities measured
by
physicists in a certain system of inertia 8.
some may be so-called field quantities
A,B,...
3595.60
u1
Among
the quantities
which are functions of
FOV H-DIMKNSIOXAL FORMULATION OF
98
35
IV,
the space-time coordinates (x in the system S. Thus a law of nature
can be expressed by one or several equations of the form
t
**} =
,...,**
0**
<>*,
(21)
<>,
a function of the quantities A, Z?,... and possibly of their
derivatives of arbitrarily high order with respect to the space-time
where
is
coordinates.
Physicists in another system of inertia 8' will now by means of the
same measuring methods generally find other values A' B' ,... for the
physical quantities mentioned, and if, for example, A is a field variable,
',
A'
will generally
of (x
).
be a different function of the coordinates
(jr't )
than
is
Thus the field variables as functions of the space-time coordinates
are generally not form -invariant.
However, the physical law expressed by
by equations of the form
(21) in
S can in
S' be expressed
A BB
'
',H',...^
t
,
;,. .)=,0,
^'L
I
(21')
where the function F in (21') on account of the special principle of relativity must be the same function of the arguments (A', #',...) as is the
function F of (A, B,...) in (21), i.e. any relation between physical quantities must be expressed by means of form -invariant or covariant
equations.
When the problem arose of expressing the fundamental equations in
the theories of electrodynamics and elasticity in a form independent of
the Cartesian system of coordinates applied in the description, the
three-dimensional vector and tensor calculus was invented. Since the
Lorentz transformations represent rotations in
(3 +- l)-space, it is there-
meet the requirement of covariance of the
attempt
laws of nature under these transformations by a generalization of the
three-dimensional vectors and tensors to four dimensions and to write
the fundamental equations in the form of four-dimensional tensor
fore natural to
to
equations.
As we
shall see in the following chapters, this
was possible
for all
fundamental equations in classical macroscopic physics, and for some
time it was even believed that all laws of nature could be written in
tensor form. With Dirac's quantum mechanical theory of the electron*)*
became clear, however, that it is necessary for certain physical systems
to deal with other quantities besides tensors, the so-called spinors j which
it
t P. A.
chap.
xi.
M. Dime, The Principles of Quantum Mechanic* (3rd
ol.,
Oxford
19-47),
IV,
THE THEORY OK KKLATIVITY
35
90
have quite different transformation properties, but nevertheless satisfy
co variant differential equations of the type of (21), (21').
36.
The four-dimensional
element or interval.
line
Four-
vectors
Consider two neighbouring points P and P' in (3+l)-space with the
coordinates (xj and (x^dx^ in an arbitrary system of coordinates S.
On account of (9) the square of the four-dimensional distance between
these points
is
given by
This expression for the line element or the interval defines the geometry
in (S-fl)-space. The infinitesimal line connecting P and P' is now the
prototype of a vector, just as in a three-dimensional space. This vector is
four 'components' (dx { ) relative to an arbitrary system of
coordinates S. By a rotation of the system of coordinates leading to the
defined
by
its
system S' these components transform
dx
Now, a four-vector
is
'i
=2
like the coordinates,
i.e.
we have
ik <fak-
quite generally defined as a quantity
which
23 )
relative
every system of coordinates has four components (a t-) which are transformed in the same way as coordinates (x^. Thus, to the rotation (3), (13)
to
of the system of coordinates correspond the transformation equations
ak
***>
for the
magnitude
(4) it
24 )
then follows from (11) and (14) that the 'square
the norm of the vector
of the four-vector', or
?
is
X-*
components of a four- vector.
In analogy with
of the
=2
!?
(25)
an invariant.
According as the invariant (25) is less than zero, equal to zero, or larger
than zero, we speak of a time-like vector, a zero vector, and a space-like,
vector, respectively.
For the infinitesimal vector (dxj connecting the two neighbouring
and P', (25) becomes identical with (22), viz.
events
cfo*
dxt
d(7
-c 2
eft
2
,
(26)
where do is the spatial distance between the two space points in physical
in the time
space in which the two events occur, while dt is the difference
of occurrence of the events.
0. If P'
the equation ds 2
The light cone with
lies
on the
P as centre is defined by
light cone, the
two events can be
FOUR-DIMENSIONAL FORMULATION OF
100
36
IV,
connected by means of a light signal. If the vector (dx^ is time-like,
P' lies inside the light cone, while it lies outside the cone when (dxj is
space-like.
From two four- vectors with the components
a
new vector with the components
(a
and
(6 t )
can be formed
(a^-f-ftj.
Since, furthermore, the invariant square of the
magnitude of
this
vector can be written
+M
ii
-2
it
follows immediately that the quantity
(27)
is
also
an invariant
in
two
scalar product of the
to each other
The
when
analogy with
(10).
Two
vectors.
The quantity
(27) is called the
vectors are said to be orthogonal
their scalar product
zero.
is
three components of a four- vector behave under spatial rotations like the components of an ordinary spatial vector a. Therefore, in
first
any system of inertia we can
temporal part
split a four- vector into
(,)
but this splitting
is
a spatial part and a
(a,a 4 ),
(28)
of course not invariant under Lorentz transfor-
mations. As the quantities
(a,
4)
transform in the same
way
as
we have obviously
for an arbitrary Lorentz transformation without
to
rotation, according
(II. 25'),
a'
+ V-
"'"
'
'
(29)
When
the vector
is
time-like,
i.e.
2>?=
|a|2-|a 4 |2<0,
(30o)
<
we can always introduce a system
vector a!
in this system.
of coordinates S' such that the spatial
We haye simply to choose the time axis
in 8' in the direction of the four- vector (a
space-like,
^a* =
|a
*-\atf
).
If,
>
0,
however, the vector
is
(306)
we can
find a system S' in
which a 4
0.
This obviously means that the
IV,
THE THEORY OF RELATIVITY
36
new time
axis is perpendicular to the vector (a^),
vector in the direction of the new time axis, fc[
at 6t
37.
a;6J
101
for, if (b T )
for
denotes a
1, 2, 3, i.e.
0.
Four -velocity and acceleration. Wave -number
vector. Four-
ray velocity
Let us
now
consider the motion of a material particle and the corre(8) in the (3+ 1 )-space. We can also use a parameter
sponding time track
representation for the time track
*t
('=
^()
1,2,3,4)
(31)
with the length s of the curve as parameter. Two neighbouring event
points on this curve are connected by an infinitesimal four- vector (dxj
with the length ds given by (22) or (26). Since the velocity of a particle
2
is always less than c, it follows from (26) that ds is negative everywhere
on the curve. It is therefore convenient to introduce instead of s a new
real
parameter r defined by
Using
(32) in (26)
_
-
1CT
we obtain
c2
dr 2
da 2 -c 2
and, since the velocity of the particle u
dr
is
dt 2
(33)
equal to dv/dt,
= (Iu /c*)*dt.
we
get
(34)
Thus r is identical with the proper time of the particle, i.e. with the time
measured by a standard clock which follows the motion of the particle.
The proper time of a particle is a measure of the length of the time track.
The time track corresponding to a uniform straight motion is repre-
sented by a straight line as, for example, the line t in Fig. 12, which
connects the two events A and B. The length of this line is given by
T>
<'--'VHJ)'
(35)
where u^ is the velocity in the uniform motion and t B tA is the difference
in time between the two events A and B. If we consider another arbitrary
time track N29 connecting the same events A and B, it obviously represents a non-uniform motion. The length of this curve is then, according
to (34), given
by
(36)
Since the expressions (35) and (36) hold in any system of coordinates
can, for example, perform the calculation in a system of coordinates
we
FOUR-DIMENSIONAL FORMULATION OF
102
37
IV,
time track v This means only
of inertia following the uniformly moving
-= 0, while the
velocity u in
particle. In this system of coordinates u l
the non-uniform motion corresponding to 2 cannot possibly be zero along
whose time axis is
that
parallel to the straight
we introduce a system
the whole track. Therefore
we must have
*:-<
1C
s1
.
(37)
1C
The straight motion is thus characterized by the fact that the length of
the time track in this case has a stationary value, viz. a maximum, compared with all other possible motions connecting the same two events.
Since the coordinate increments
on the time track of a
and dr is an invariant,
(r/.rj
are the components of a four- vector
particle
be components of a four- vector which is called the four -velocity.
(38) and (34) we see that the components of U are
will also
From
where
is
(40)
the usual three-dimensional velocity vector. The four-velocity is a
lies in the direction of the tangent of the time track
four-vector which
and the norm of
one obtains
Thus the
c2
this vector has the constant value
four- velocity
by differentiation of
for
from
(39)
.,
is
(4 1)
a time-like vector of constant magnitude and
we
get
If we apply the equation (29) to the four-vector
come back to the transformation equation (II. 55')
the equation inverse to
The four-vector
(II. 56).
given by (39)
for velocities
and
we
to
IV,
is
THE THEORY OF RELATIVITY
37
called the four-acceleration
and from
(39)
we
get for
103
its
components
(42)
where
du
-
r/
-x
2
rf/
the usual three-dimensional acceleration of the particle. According
to (41') the vectors U and A are orthogonal to each other. In the rest
is
8' of the particle the
system
components of A are
t
^'.-(a',0).
(42')
Another example of a four-vector is given by the wave number vector
of a plane monochromatic wave The invariant phase F of the wave
(cr,)
in an arbitrary system of coordinates 8 can, on account of (II. 68), be
written
Here, n denotes a unit three- vector in the direction of the wave normal,
the frequency ?r is the phase velocity, and A is the wave-length of
v is
the wave.
As the phase
an invariant we have, according to
is
,A
and, since this equation
which shows that
(3),
must hold
for arbitrary
x k we get
,
a four- vector. Equations (44), or their equivalent
23 for the
(29), yield directly formulae (70), (71), arid (72) deduced in
transformation of the characteristics of a plane wave.
(a,) is
Chapter II, we saw that the ray velocity in a medium with
refractive index n > 1 transforms like the velocity of a particle with the
In
24,
velocity u'
with (39)
cjn in the rest system of the medium. In analogy
therefore define a four-ray velocity vector (f/J with the
w' -
we can
components
where u
is the magnitude of the ray velocity and e is a unit three-vector
in the direction of the ray.
In the rest system $' of the refractive medium where e'
n' and
w'
u'
c/n,
we have
cri
icn
(46)
FOUR-DIMENSIONAL FORMULATION OF
104
In another arbitrary system of inertia S we then have
From
--
U\oL tk
the two four-vectors
and
(45)
o-
5X
"',
(47)
<>.
>
(43)
Uk
(46) it follows that
SX ",
i.e.
37
IV,
and
f^
are orthogonal to each other. Inserting
we obtain
or
w;
(u.n).
(47')
(47) thus expresses simply that the phase velocity is equal to the projection of the ray velocity in the direction of the wave normal in any
system of inertia.
While the square of the magnitude of (U
we have
is
c2
constant, equal to
Thus, for a light wave in empty space the phase wave vector
is
a zero
vector.
Four -momentum. Four-force. Fundamental equations of
point mechanics in four -dimensional vector form
In 28, formulae (32) and (37), we have seen that momentum and
38.
energy
(p, K[c) of a material particle
transform in the same
way
as the
Therefore the four quantities
space-time coordinates (x,r).
Pi-- (p,iM/c)
(49)
are the components of a four- vector, the four-momentum vector. When
we use (III. 23) and (III. 31) in (49) we see that the four-momentum
proportional to the four-velocity (39) and the factor of proportionof the particle, i.e.
ality is equal to the rest mass
is
p,
On
account of
(50), (49),
IP?
and
-m^.
(41) the
= P*---* -
norm
2
<
in accordance with (III. 35)
(50)
f;f
of this vector is
--
-8<- a
(< >i)
The four-momentum vector
is
thus a time-
like vector.
Since p in (50) and a, in (43) are transformed in the same way, it is
possible in ari invariant way to adjoin a plane wave with the wave
t
number vector
cr
to a free particle with
four-momentum p^ such that
K,
(52)
/,
=-
THE THEORY OF RELATIVITY
38
IV,
where h
105
an invariant constant. If h is chosen equal to Planck's conwave is the de Broglie wave| of the particle. The
w
of
the
de Broglie wave is, according to (43), (52), and
phase velocity
(III. 36), connected with the velocity u of the particle by the equation
is
stant, the adjoined
-_-
w
The theorems
*l
J.
cor 4
|P|
p-
=.-
cp 4
of conservation of
U
(52')
.
'
r2
momentum and
energy in a collision
between particles can now be comprised in the equation
where p^ and p are the sum of the four-momentum vectors of the particles before and after the collision, respectively. The first three equations
--(i
1,2,3) represent the theorem of conservation of momentum while the
t
fourth equation, corresponding to
-- 4,
expresses
the conservation of
energy.
In (53) these theorems are expressed in a form in which the co variance
under Lorentz transformations is obvious, for two four-vectors whose
components are equal in one system of coordinates will obviously have
equal components in any system of coordinates.
As mentioned in the conclusion of 28, the quantities (F^ 7 (Fy.u))
transform in the same way as (p, E), i.e. the four quantities
,
are the components of a four- vector, Mirikowski's four-force. \ The
fundamental equations of mechanics (III. 42) can thus be written in the
following co variant four- vector form
or, since
is
constant in time,
ro
^-F
dr
The
(56)
three equations are the equations of motion, and the fourth
equation expresses the theorem of conservation of energy. Conversely,
from the validity of (55) in every system of inertia it follows at once that
first
is a four-vector, for
must then transform in the same way as the
left-hand side of (55) which is obviously a four- vector, since r is invariant.
f\
Op
H. Mmkowski,
cit
chap, u,
k
p.
58
Das Relativitatsprmzip', Ann.
d.
Phys. 47, 927 (1915).
FOUR-DIMENSIONAL FORMULATION OF
106
From
(41')
and
(56) it is seen that the vectors L\
to each other, viz.
yp
, r
.._
and F are orthogonal
l
ir^\
\
'
38
IV,
This property of Fl is closely connected with the constancy in time of
the rest mass. From (55), (41), and (41') we obtain
yp
jjl
__
X^
J-*
[jl
fl(
o *-i)
__
dr
jjzl
<^-4
^m o
dr
j
'
^C
^-4
jj d^>i __
l
^2
_Q
dr
dr
<
(58)
Equation (57) thus expresses the fact that the proper mass
mQ
is
con-
served.
Sometimes we are concerned with systems
in
which the external
forces produce a change in the proper mass of the particle. This is the
case, for instance, in an electrically conducting substance under the
influence of electromagnetic forces, since the Joule heat energy produced
in the body will contribute to the proper mass. If in such a case we wish
to maintain the equations (55), we must in the expression (54) for Fx
A by the total effect. While Fv F2 F3 remain unchanged,
replace F u
F* will be defined by
.
- ^
*\
A ^
where
(F.u)
'(60)
the mechanical work performed per unit of time, while Q is the amount
of heat or non-mechanical energy developed per unit of time in the body.
The fourth equation (55) which, by means of (49), (59), and (34) can be
is
written
=^+.
(61)
thus again expresses the conservation of energy. As the equations (55)
in every system of inertia, and since the left-hand side is a
must hold
four-vector, the generalized four-force must also be a four-vector.
now (FL ) is no longer orthogonal to (f/,). Instead, we have
=__
Since the left-hand side of this equation
must thus be an
in the rest
invariant. It
is
system per unit time.
is
9 _
'
But
(62)
^
>
an invariant,
equal to the
amount of heat developed
IV,
THE THEORY OF RELATIVITY
38
107
In a physical system of this kind the rest mass of the particle
From
longer conserved.
and
(58), (62),
(tWln
-7-
no
get
/f*A\
Vj/
64
c*
identical with the time in the rest system, (64) simply means
amount of heat developed in the rest system has an inertial mass
is
that the
corresponding to Einstein's equation (III. 74).
While the equation (55) thus holds quite generally, (56)
when the. rest mass is conserved, i.e. when
The equation (55) may also be written
m
on account of
valid only
(58),
^F
is
== 0.
dr
dr
Thus, when
^FU
dU,,dm
0u = f
dr
or,
is
dr
Since r
we
(63)
f/,
0, i.e.
c*
when
the
1'
proper mass is not conserved, a force
maintain a uniform motion of the
will in general he necessary in order to
particle; for
dUJdr
only
if
I
tf
IT \
-Jd
(JO
u = v Uif
i
In the rest system this means
0,
(65)
but in every other system of inertia
we have
From
amount
(63)
we get at the same time the transformation properties of the
of heat
plying (63) by
Q conveyed to a system in a process
Ar^l
u 2 /c*) we
of that kind. Multi-
obtain, using (34),
Q A is the total amount of heat conveyed to the system during
an interval A, while A$ = Q Q AT is the corresponding quantity
measured in the rest system. Thus we have
In Chapter VII we shall see that the equation (66) also holds for any
amount of heat conveyed to a system in a thermodynamical process.
FOUR- DIMENSIONAL FORMULATION OF
108
39.
Tensors
of
39
IV,
rank 2
In the preceding chapters we have seen that the covariance of the
fundamental equations of mechanics under Lorentz transformations can
be expressed in an especially elegant way by writing them in fourdimensional vector form. To obtain a similar geometrical representation
of electrodynamics, for instance,
dimensional tensors also.
By a
tensor of second rank in
has 4 2 components
(t lk )
it
necessary to introduce four-
is
(3+ l)-space we mean a quantity which
relative to
an arbitrary system of coordinates S
another arbitrary system 8 are con(t lk ) by means of the equations
and whose components (t\ k
nected with the components
in
where the a lk are the same
the transition from
coefficients as in
equation
(3)
67 )
which defines
to S'.
For the sake of simplicity we have here omitted the sign of summation,
substituting it in the following by the convention that an expression in
which a Latin index, like / or m in (67), appears twice shall be summed
over this index from 1 to 4. Free indices like i and k in (67) can assume
independently the values 1, 2,
in the future be written a l a l
values
1, 2,
3 it will
Equation (25), for example, will thus
a\a\. If an index can assume only the
be denoted by a Greek letter, and if it appears twice
3, 4.
an expression it is implied that we shall sum over this index from 1 to
The square of the magnitude of a spatial vector will thus be written
in
a|
The
a a
t
definition (67) of a four-tensor
3.
is
a direct generalization of an
ordinary spatial tensor whose components by rotations in physical space
transform according to the equations
where
LK are the coefficients in the orthogonal transformation reprethe
rotation.
senting
The sum of the diagonal elements of a tensor of second rank is an
OC
invariant, for from (67)
In the same
way it is
an invariant.
(11)
we obtain
means of the orthogonality
seen by
(14) that the quantity
is
and
ik l ik
__
.,
.,
hmhm
relations (11),
IV,
THE THEORY OF RELATIVITY
39
If a Lk represents
space,
r
and
,a
simple rotation in the three-dimensional physical
we have
(67)
is
109
reduced to
C=
aiA<V*A/i
*'i4
== a
^4*
*^4/i
a <A^A4
(72)
^44
^44
'
shows that the spatial part of a four-tensor by spatial rotations
like an ordinary three-dimensional tensor. Furthermore, the
numbers t4 and t K separately form the components of a spatial vector,
while / 44 is an invariant for purely spatial rotations.
From a vector a and a tensor t lk a new vector can be formed whose
(72)
behaves
in
components
every system of coordinates are given by the equation
*,
for
from
If a lk
(24), (67),
and
viously also
and
(II)
ik
a k9
(73)
we obtain
components of two tensors, fl^+^
be the components of a tensor which is called the sum of the two
b lk are the
a tensor, t tk
t
kl is also a tensor, viz. the transa
b
denote
tensor.
When
and
the
posed
components of two vectors
which thus are transformed according to (24), the quantities
tensors. Further, if
lk is
t
',*---
will obviously
The tensor of second
product of the vectors a and b k
be transformed according to
rank, defined by (75),
Also the M
quantities
is
called the direct
*ik
a b k~ a k b
i
75 )
(67).
,-.
76 )
*ki>
(75) and its transposed,
tensor like (76), satisfying the equation
which denote the difference between the tensor
thus represent a tensor.
for all values of the indices
and
/*,
is
called antisymmetiical.
Analo-
gously, a tensor satisfying the equations
',*
'*i-i,A
78 )
called symmetrical.
Since both sides of the equations (77) and (78) transform like tensors,
these equations must hold in every system of coordinates if they hold in
is
one system. Symmetry or antisymmetry of a tensor
is
thus an invariant
FOUR. DIMENSIONAL FORMULATION OF
110
IV,
39
property. In an antisymmetrical tensor all diagonal elements are zero,
for if we put i
k in (77) (without summing), we obtain
^for
any
An
six
(79)
antisymmetrical four-tensor of second order
Fik =
Fkl
has only
independent components. Putting
H tK
-*,*
i.
E =
FiK9
= -tT
iF ,
(80)
4l ,
according to (72), in spatial rotations (71) behave like
of
an
components
antisymmetrical spatial tensor and an ordinary space
vector E, respectively.
IK
and
will,
Moreover, putting
//!
#,=
#*,
//,
#3 =#12,
we obtain
the following transformation equation for
of a special Lorentz transformation (16),
(80')
H and E in the case
t
=E
19
'
E'2
The corresponding reciprocal equations are
obtained by interchanging the primed and unprimed quantities and
v.
With v = (v, 0, 0) the latter equations may be
replacing v by
where y
(1
v 2 /c 2 )-*-.
comprised in the vector formulae
E=
yE'-f:
H=
and
in this
(81')
yH' + -i
v2
form they are valid
for
any Lorentz transformation without
rotation of the spatial axes.
40.
Angular
momentum and moment
of force in four -dimen-
sional representation
Let (#J
(x,ict) be the space-time coordinates of an event point on
the time track of a material particle, and let (p t )
{p, i(E/c)} be the four-
momentum of the particle.
According to
we can form an antisymmetrical
J^f
The
__
spatial part of this tensor
is
(76),
from these two four- vectors
tensor
^ p
y.
rp
(82)
an antisymmetrical space tensor, the
THK THKORY OF RELATIVITY
40
IV,
111
whose components are connected with
LK
angular momentum tensor
the angular momentum vector
,
M
bv the equations
In
M=
(Mf
M,,,
the same way we can from
xx p
= (Mn
s)
(x,)
(83)
M Mn
3l ,
(84)
).
and from the Minkowski four-force
^ ^ ^ ^_^ ^
(FJ form the tensor
(85)
The
spatial part of this tensor gives, analogously to (83), (84), the
moment of the Minkowski force (III. 41) relative to the origin.
By means
of the fundamental equations of mechanics (55)
we
obtain,
using (38),
or,
by means of
(50)
and
(85),
dM
lk
(86)
The
spatial part of this equation contains the angular
theorem
momentum
^=
(xxF).
(86')
The Kronecker symbol defined by (12) represents a tensor of second
rank of especially simple character. Consider a tensor whose components
in 8 are equal to S, A on account of (67) and (14) its components in $' are
;
'
This tensor has thus the same constant components in every system of
coordinates and it is the only tensor whose components remain un-
changed by a transition to another system of coordinates.
^~~ ^
41.)
/
^
Tensors of arbitrary rank
In analogy with (67) a tensor of third rank in (3-fl)-space is now
defined as a quantity with 4 3 components t lkl which transform according
to the equations
C/
a 7m
(X-kn <*/;>
4>/^>'
^kl
*row;>
a mt <*nk a j)l'
88 )
Thus every index transforms separately according to the same law as
for a four-vector.
A $owr-vw^
same sense an
aiajUA^aJe.nsor^)
tensor of rank n is then a quantity t ikl ...
rank.
In
the
zero rank.
each of whicTT transforms separately according to the law
(24),
FOUR-DIMENSIONAL FORMULATION OF
112
IV,
41
characteristic of a vector. If in a tensor of rank n two indices, for example
i, are put equal to each other, we obtain after summing over this
index a tensor t lllm of rank (n 2). This is a direct consequence of the
k and
transformation equations for tensors in connexion with the orthogonality relations (11), (14). Such a process by which from a tensor of rank
n a tensor of rank
in
2) is
formed
is
called contraction.
Equation (69),
which a tensor of rank zero has been formed from a tensor of rank
(n
represents a special case of such a contraction.
By addition (or subtraction) of corresponding components of two
tensors of rank n we naturally obtain a new tensor of rank n. On the
2,
other hand, it has no covariant meaning to add two tensors of different
rank. However, we can always form the direct product of two tensors of
ranks n and m, respectively, by forming all possible products of the
components of these
tensors.
The equation
Hereby we obtain a new tensor of rank
a special case of this
a
the
t
b
of
since
tensor
rank 2 is the direct
l
k
general theorem,
lk
a
of
first
rank
and
b
of
the
tensors
two
l
k
By subsequent conproduct
(n-\-m).
(75) obviously represents
traction of the tensor d b k
l
we obtain
a tensor of rank zero, viz. the
invariant (27).
Equation
(73) also represents a special case of a
combination of both
operations: direct multiplication and contraction. Primarily, a tensor
(t ik .a ) of rank 3 is formed by direct multiplication of the two tensors
t
lk
and a
contraction
By
we then obtain the
tensor b l
lk
a k of
first
same way, equation (70) can be regarded as a result of a
multiplication of t lk by itself succeeded by two contractions.
rank. In the
direct
42.
Pseudo- tensors
In three-dimensional vector calculus one introduces besides the
ordinary (polar) vectors with the transformation law
a
't
so-called axial vectors
<***
89 )
which transform according to the equations
a\
ococ
lK
a^
(90)
where a = \oc iK is the transformation determinant. For proper rotations
we have a = 1, and an axial vector transforms like a polar vector.
However, by reflections in which one or three axes change their signs we
1
have a
thus, for instance, by a reflection at the origin in which
\
x[
- -x
(91)
the components of an axial vector are unchanged, while the components
THK THEORY OF RELATIVITY
42
IV,
113
of a polar vector change their signs. An example of an axial vector is the
vector product c of two polar vectors a and b with the components
c
---
(c^Cg.rg)
((I 2
a 1 b^a 1 b 2 ~a 2 b l ).
b^-a 3 b 2 ,a 3 b l
(92)
= fo we have obviously c[ C Another
b[
By a reflection a[ =
well-known example of an axial vector is the magnetic field vector H.
The generalization of the notion of an axial vector to tensors of higher
rank and to four dimensions is obvious. These quantities are called
pseudo-tensors. They transform like tensors, except that they are also
t
defined by (15).
multiplied by the transformation determinant at
\oc lk
2
2
a
of
4
rank
a
is
with
Thus, pseudo-tensor
quantity
components in every
\
system of coordinates with the transformation law
t\ k
----
oux lt
ot
km ti m
(93)
we get at once the following rules. The sum of two
the
of
same rank is again a pseudo-tensor of equal rank.
pseudo-tensors
of
a pseudo-tensor and a tensor is a pseudo-tensor
The direct product
From
this definition
with a rank equal to the
sum of the ranks of the two factors in the product.
two pseudo-tensors is a tensor. The operation
of contraction can be performed with pseudo-tensors in the same way
as with tensors, thus leading to a pseudo-tensor whose rank is diminished
The
by
43.
direct product of
2.
The
Levi-Civita symbol
Like the Kroiiecker symbol, which was shown in 40 to be a tensor
with constant components in every coordinate system, the LeviCivita symbol is a pseudo-tensor with the same property. In fourdimensional space this pseudo-tensor is of rank 4. The Levi-Civita
symbol is defined as a quantity S, A/m which is antisymmetric in all four
indices.
which
Thus,
the only
won -vanishing components of
all four indices are, different
and they are equal to
8 lkhn are those for
1
or
according
an eren or an odd permutation of (1,2, 3, 4). Now consider
(i, A*, /, m)
a pseudo-tensor which in 8 has the components 8 <A/m In another system
as
is
S' its
components are then
mr 8 w/f , r
Since
all
8'tUm is
symmetry
3595.60
(i,
A%
(94)
by the transformation,
and we need only calculate the
for which -we get
properties are conserved
also antisymmetric in all indices
component with
= (1,2,3, 4)
&'m4 = *i,,2,,3v*4A,,,,.
/,
m)
(5)
FOUR-DIMENSIONAL FORMULATION OF
114
From
the definition of 8 lklm
by means of
thus,
and
(95)
follows that
it
(15'),
8 1234
From
43
IV,
<*
==
S 1234-
the symmetry properties of 8 lklm and 8 lWm
8 !*/m
it
then follows that
8 tA/m
96 )
(i, i% /, m), which shows that the Levi-Civita
a
with
the same constant components in every
pseudo-tensor
symbol
coordinate system.
for all values of the indices
is
In three-dimensional space, the Levi-Civita symbol
is a quantity
equal to I, the other nonvanishing components following from this by the symmetry rules. It is
shown in the same way as before that 8 llf^ is a three-dimensional pseudo-
8 lK x
antisymmetric in
all
three indices. S 123
is
tensor.
Dual tensors
By means of the Levi-Civita symbol we can
44.
metric three-tensor
LK
associate
with a pseudo-vector (axial vector)
an antisymH by
#i=4SiKAtf*A,
H-
i.e.
(H19 #2
ff8 )
(#23
(97)
#31,
(98)
12 ).
The quantities l defined by (80') are thus the components of an axial
vector dual to the tensor IK
If
is
IK
of the form
",
= A,-aA=
(99)
I'
where a and b K are two vectors, the corresponding axial vector
t
^ = S^a^A
(100)
the vector product o
a x b. This tensor or its dual axial vector
represents the parallelogram formed by the vectors a and b, their components being equal to the projections of the parallelogram on the three
is
The area a of the parallelogram
coordinate planes.
equation
M
<r
The pseudo-vector
(100)
we
<r
get
*ti
is
^1=
given by the
/im\
4< T K< r
is
(101)
c-
perpendicular to the parallelogram, for from
SHCA^A
and
a b
i
(102)
IV,
THE THEORY OF RELATIVITY
44
115
on account of the antisymmetry of the Levi-Civita symbol. The vector
dual to the tensor
IK
fa K
dx
__ fa;
'dx
similarly the axial vector curl a.
Three vectors a, b, and c define a parallelepiped which by analogy
with (99) is represented by an antisymmetrical tensor
is
(103)
By means
of the Levi-Civita symbol we can associate this tensor with a
pseudo-tensor of rank zero, i.e. with a pseudo-invariant
v
F
ftl
=~
a2
represents the volume of the parallelepiped.
proper rotations, but changes sign by
F2
'
(104)
^2
It is invariant for
We
reflections.
obviously have
V
'
(105)
If a, b, c are infinitesimal vectors
lying in the directions of the x^,
x 2 - and x 3 -axes, respectively, for instance
y
(efcq, 0, 0),
the corresponding volume element
dV
(0,rf# 2 0),
is,
by
(104)
and
dx^x^dx^.
(0,
0,d^ 3 ),
(106)
(106),
(107)
a pseudo -in variant.
In (3+l)-space we can associate an antisymmetrical tensor of rank
n
4 with a pseudo-tensor of rank (4 r?)
by means of the Levi-Civita
It
is
<
symbol
tensor
8 iklm
Flk is
Thus the dual pseudo-tensor
defined by the equations
F*k
^um f
lm
FJk
to
an antisymmetrical
(108)
i.e.
1
= ^.
(109)
^ 23'
denned by (80) and (80') we see that
Introducing the quantities E and
the dual pseudo-tensor Ffk is obtained from F
lk by the substitution
FOUR-DIMENSIONAL FORMULATION OF
116
E ->
H ->
H,
The equations
E.
and
(81)
(81') are
44
IV,
unchanged by
this
substitution, in accordance with the fact that the Lorentz transformation
without rotation
is
a transformation with a
same way
case transforms in the
tensor
as
lk
1,
so that
F*k
in this
of the special form
a^ k
where a and b k are two vectors, represents the two-dimensional 'parallelogram' defined by the vectors a and b by analogy with (99). The dual
l
tensor
*.
is
orthogonal to the vectors a and b k and to the tensor alk
^-^A-<4<^-<>.
The area a of the parallelogram
is
(HI)
defined by
by analogy with (101).
Likewise, to an antisymmetrical tensor of rank
pseudo-tensor of rank
1,
e.
3 corresponds a dual
a pseudo-vector. If the tensor is of the form
(113)
"
''
*/
where
b^
are three independent vectors, the dual pseudo-vector
c,
where
(iklm)
Vlkl and V
is
an even permutation of
ft
t ,
c t-.
is
The volume V of the
V
l
(1234).
orthogonal to this space, for
JX = FA =
vector
represent the three-dimensional parallelepiped defined
the vectors a p
parallelepiped
is
is
Fc
t
0.
we
by
get from (114)
(116)
given by the length of the pseudo,
F*=-F F = lFWm F
t
fc/m
(117)
THE THEORY OF RELATIVITY
44
IV,
117
Finally, the dual pseudo-tensor of an antisymmetrical tensor of rank 4
a pseudo -invariant. If the tensor is of the form
is
(118)
am
the dual pseudo-invariant
v
-
dm
Cm
is
__ - g
__
f
iklm a b'k cl d m
S,
bm
(119)
represents the volume of the parallelepiped defined by the vectors
abcd
We
have
(120)
If a p b l c
,
e ,
d are infinitesimal vectors lying in the directions of the co4
respectively, the corret
ordinate axes and of lengths dx dx 2 r/a* 3
sponding four-dimensional volume element
}
is
(121)
which
is
thus a pseudo-invariant.
45. Infinitesimal Lorentz transformations. Lorentz
transforma-
tions without rotation
An
infinitesimal
the form
where the
we
in
linear transformation (x z ) ->
homogeneous
x^
-x+
,
lk
xk
/s
lle
has
/ioo\
(122)
(B lk
(x\)
)x k9
are infinitesimal quantities. For a Lorentz transformation
get, using (122) in (10) and neglecting terms of second order
* lk
*>
Since this equation must hold for
all
we must have
values of x l and x i
t*=-^
'
This condition
is equivalent to the orthogonality relations
the case of an infinitesimal transformation.
Now consider an arbitrary Lorentz transformation (3)
space-time coordinates of two systems
S and
S'.
Let v
123 )
(11), (14) in
connecting the
(vx v u vz ) be
,
FOUR-DIMENSIONAL FORMULATION OF
118
IV,
the velocity of 8' relative to S. The corresponding four-velocity
by
(39)
F =(yv,yic)
The components of this
(l-/c)-').
(0,0,0,
ic),
the four- velocity of a point at rest in S', which means that the
the direction of the a^-axis. From the transformation
J^ is
vector
V is then
four- vector relative to S' are
F;=
since
(y
45
lies in
we then
equations of a four- vector
Vk
z
get
V\oi lk
3)
^ are unit vectors
Similarly,
#3-axes, respectively, we have
if e( \ e( \
#'
and
(.)
__
___
__
~~
~~(o
<o
in the directions of the x\-, #'2 ~>
(t=l,2,3)
8*.
d)>
(124)
ICOL^.
In the case of a Lorentz transformation without rotation the coefficients
oL
lk
follow from
V
(II. 27):
-.v
"
--i(y-i)
^ (y-1)
(125)
i(y-i)
In (3+l)-space it represents a rotation in the two-dimensional plane
defined by the time axes of S' and S.
If v x v u vz are infinitesimal quantities, we have, neglecting second,
order terms in these quantities,
and
(125) reduces to
a I/,
& t&
lk
+
I
lk
tfc
with
IK
lie
0,
>
l4
(4
4l
41
^>,
c
44
0.
(126);
V
46. Successive Lorentz transformations
Let
*;
* |A * A
8-+S',
xl^oc^xl
be two successive Lorentz transformations.
formation
W
r"~-U',v
X
(a^oiiLJXL
t
S'^S"
The resultant
(127)
trans-
H9^
-8
V
THE THEORY OF RELATIVITY
46
IV,
of course, again a Lorentz transformation,
is,
same orthogonality
satisfy the
i.e.
relations (11)
119
the coefficients
and
(14) as
oc
lk
and
oL
lk
not in general represent a Lorentz transformation
without rotation even if this is the case for the two transformations ( 1 27),
(128) will
However,
i.e.
Qt!'
lk
will
not take the form (125) even if ot ik and oL ik are of this form.
if the three time axes in S, $', and S" are lying
This will be the case only
in the same plane.
In the special case where the transformation from S to S" is an infinitesimal Lorentz transformation without rotation, we have, according
to (126),
*ik
8 t*
*IK
'ifc
= ~
't4
4t
44
>
129 )
V( being the four- velocity of S" relative to S'. Hence,
<*lk
8 t/+ </)/*
Let
xi
(1.30)
*ik+*il<xik'
= f (r)
t
represent the time track of a particle in arbitrary motion in S, r being
the proper time of the particle. We shall now try to determine the
successive rest systems of the particle such that two consecutive rest
systems at any time have the same orientation of the spatial axes. Let
S'
and S"
in (127) be
momentary rest systems of the particle at the times
The four- velocity of S' relative to S is then
r and T-fdr, respectively.
^(T)
Similarly, the four-velocity
tft (T)+dtft (T)
= U (T)+tTt dr = /<(T)+/(T) dr.
7;(T)
V't
U:+dU't
(132)
The components of these two
(131)
^=/,(T).
of S" relative to S is
four-velocities in S' are
= tt t^ = (0,0,0,tc),
= a ik (Uk +dUk = U'i+a^ dUk
]
(133)
(134)
to S" was supposed to be an
mal Lorentz transformation without rotation, the coefficients
Since the transformation from
transformation from
(129) being given
by
to S" are obtained from (130)
Now,
(134).
obviously,
and
infinitesin
oLik
in the
(129), V\ in
we have
C
for,
on account of (133),
(129).
this is seen to
be identical with the
t'ik
defined in
KOUR-DIMENSIONAL FORMULATION OF
120
46
IV,
Thus, from (130), (135), and (134)
ij\du'-ir dv\
=
,*-* =
-ir~~- <%
The
coefficients
then equal to
oi
oc
lk
can
IT
...
..
-,a( r7k dUt-Ui
JTr
dU
k
).
now be regarded as functions lk (r) of r, a"lk being
Hence we get the following differential equa<y.
lk (r~\-dr).
tions for the functions a lk (r):
(136)
..,
with
ri %k
For later use we note that the
U^lJi.
C/y
-J-A_JL_J
C/T.
coefficients
Kl
ot
^l
lk
/,
and
rj
lk
satisfy the relations
.=
on account of
(14), (41), (41')
and
(137)
33 X
(133).
The equations (136) determine the transformation from the fixed
S'(T) with coordinates
system S to the momentary rest system #'
x' thus we have
^ =
/ion\
'
\
x k oi kl (r),
/
(139)
or, if we
make a continuous displacement of the origin in
particle
is
S' such that the
always lying at the origin of the rest systems $'( T )>
(140)
).
now
attach a space vector e'(r) of unit length to the particle
considered in such a way that the components e' with respect to the
spatial axes of S'(T) have the same values at all times. e'(r) may, for
,
Let us
T/
We
thus have
instance, have the direction of one of the space axes in
at any time a displacement of e' without change of orientation. In
A$>
(3+l)-space this vector
is
represented by a space-like four- vector with
components given by
in 8'.
Its
e ; -= (e', 0)
(141)
components in S are given by
^(r)
e l is
orthogonal to
19
= fa^r) =
e'^T).
(142)
since
e t t7<==e;C/;
(143)
IV,
THE THEORY OF RELATIVITY
46
on account of
now have
and
(133)
,
_
~
dr
From
(141).
^'.
dr
and the velocity of the
we have in general e r ^
zero again,
we shall
for r
<
Cf
(144)
we have
for
moment. At a
particle is zero at that
e\,
and (143) we
(142), (136), (137),
dr
i.e.
If S'(r) coincides with
121
later
time
and even if the velocity of the particle becomes
have
in general
er
(e,0)
=<==
(e',0)
means that the components of the unit vector considered are different in 8 and AS", in accordance with the fact that the
vector has performed a Thomas precession relative to S (see 22).
at that time. This
47. Successive rest
systems of a particle in arbitrary rectilinear
in constant circular motion
motion and
Let the motion of the particle be in the direction of the o^-axis, then
we have /2
Further,
time r
/3
0, i.e.
Vt
(A,
0,
U =
0,/4 ),
0,
0,/4 ).
(145)
shall assume that the particle has zero velocity in S at the
and that S'(r) coincides with S at that time. On account of
we
= U\+U\ =
U - (csinh0(r),
U U
the equation
form
c2
we may
0,
where
(A,
(r)
\c
is
in the
(146)
zero for r
0.
Hence,
iccosh0(r)),
0,
an arbitrary function of r which
9(r) is
therefore write
sinh
dr,
0,
ic
0,
^o
cosh 6 drl
(147)
'
Further, if 4^( T ) represent unit vectors in the directions of the spatial
axes of the successive rest systems S'(r) we have
40(0)
Each of these vectors
satisfies
g fci
for
0.
the equations (144)
(148)
;
/JpW
therefore
=
^dr
(epUg)Uk /c*.
(149)
FOUR-DIMENSIONAL FORMULATION OF
122
On
IV,
47
account of (145) we see at once that
are solutions of (149).
To
find the
cumstance that
components
Qj
are integrals of the equations (149), as
(149)
by
e o )e u)
is
e (11)
and
1)
e^
we use the cir-
(151)
seen at once by multiplying
Uk and e^ respectively. Hence
0,
-c
(152)
e u>2,
i.e.
1C
(153)
From
(124)
and
(124')
we
therefore simply get
0100
0010
iUJ^
'UJic
(154)
UJic/
'JJic
which correspond to special Lorentz transformations (see (16)).
By means of (154),. (146), and (147) the transformation equations
(140)
may
be written
xl
si
Xn
(155)
Xo
T
#4
ic
cosh
dr-\-x\
sinh 0(r) -\-x cosh 0(r)
In the special case where the motion of the particle
get from (III. 46)
is
hyperbolic
we
i.e.
(156)
IV,
THE THEORY OF RELATIVITY
47
Thus, from
and
(III. 47)
123
(156)
/3
0,
(157)
sinh~-
(158)
1
comparison with (146) shows that
d(r) in this particular case is
(159)
and the transformation
*-'
(155) reduces to
i
'
cosh-
x,
g\
i
v
i-- sinh --
-4'
(160)
*'2>
tX/2
x\'
sinh -
4- x*
cosh --
of the transformation coefficients (154) we get for the comof the four acceleration vectors of the particle in the succesU[
ponents
By means
sive rest
system
/:
$'(T)
|A
?A
u* k
l~*
\
or,
on account of
fll
ic
u*, o, o,
+~u
c
(158),
#;
(?,
0,0,0),
(161)
which shows that the accelerations of the particle in the successive
rest
systems are constantly equal to g (see equation (42 )).
shall now briefly discuss the solution of the equations (136) in the
case where the particle is moving in the (x l x 2 ) -plane with constant
We
angular velocity
withy
aj
(lu*/c
in a circle of radius a. In this case
)-*,u
in the circular motion.
acu
From
we have
being the constant velocity of the particle
(162)
we
get
sin(coyr), 0,
<
FOUR-DIMENSIONAL FORMULATION OF
124
By
IV,
47
a straightforward calculation it is easily verified that the following
coefficients a lk provides a solution of the equations (136):
scheme of
1AV
cos
a:
cos
sin
coy a sin/?
a:
sin a cos j8
sin
sin
(\suijS
01
a cos/?
sin asin/J-fy cos
y sin a cos/?
y cos
i^c
!^ c
c
uy cos a
-
(164)
with
at
a>yr,
ya
j8
The transformation
AS
r.
a>y
-> S'(r)
is
then obtained from (139) or (140) by
introducing the expressions (164) for
For r
we
oc
lk
get
(165)
Thus, denoting the space-time coordinates of the system
get from (139)
^i
==
x"'
^2
yx
\i
#2
<S"(0)
by x
we
(166)
^3 "^ ^3?
3*4
==*
0:
+ y^4
which represents a special Lorentz transformation from S to a system
S'(0) moving with the velocity u in the direction of the # 2 -axis.
At the later time r -= r t -= 27r/coy we get from (164)
cos
ysin^
uy
_ l.
COS /
c
I
(167)
-0
r
where/?!
27r(y
1).
The coefficients a ^(T^) may, however, be written as
THE THEORY OF RELATIVITY
47
IV,
125
0\
ft
sin ft
sinft
cosft
(16g)
(cos
I/
If the space-time coordinates of the system S'(T}) are denoted
or
x\
cosft
x\
we
#5- sin ft x,
sinft zj+cosft
#|
by
(169)
a?g,
Thus the system S'^^) does not coincide with the system
S'(0),
but has
to be rotated through an angle ft in the (a^a^) -plane in the direction of
motion of the particle in order to give the spatial axes of ^'(TJ) the same
orientation as the axes of $'(0) or as the axes of S. In other words, the
axes of S have to be rotated through an angle
1) in the
2n(y
ft
(^ 1 x 2 )-plane in order to give them the same orientation as the axis of the
system
$(TJ).
This
is
due to the Thomas
(II. 65) for the velocity of the
T,
we
Thomas
T
get
o> dt
(y
and vT
x v in our
= vwT
1)
a constant vector perpendicular to v and v
the total angle of precession is just
case
27rv,
V
VX ~dt
v*
and, since v
Integrating the formula
precession over a whole period
effect.
is
-27r(y-l)
-ft.
(170)
Tensor and pseudo- tensor fields. Tensor analysis
As in ordinary space we speak of a tensor field of rank n in (3+ 1) -space
48.
any point in this space is connected a tensor of rank n. In particular,
we have a tensor field of rank zero, a so-called scalar field, if an invariant
number is connected with every event point. This means that we have a
if to
certain function of the coordinates
<f>(x)
<(# t-)
^(x^x^x^x^)
in
every system of coordinates S, such that
<f>'(x')
<f>(x),
(171)
the function corresponding to the system of coordinates S'
and the numbers (x[) and (X T ) are coordinates of the same event point in
if <f>'(x') is
the two systems S' and S, respectively. In general, <f>' will be a different
function of the variables (x() than
is of (x^. Thus a scalar function
<f>
^( x i)
is
generally not a form -in variant function of the coordinates. This
FOUR-DIMENSIONAL FORMULATION OF
126
will in fact
which
is
only be the case
if
<f>
48
IV,
a function merely of the quantity
is
(4)
also invariant in form.
Analogously, we have a tensor field of rank 1 when a four- vector is
connected with every event point. The components a t (x) and a((x') of
the four-vector in two arbitrary systems of coordinates 8 and S' will
then be functions of the coordinates of the event points, and
a\(x')
when the connexion between the
* lk a k (x)
(172)
variables (x
(x[)
and
(x)
given by (3) and (13).
For tensor fields of higher rank equations exactly analogous to
and
(x^
is
71
(172) are valid.
Now from an arbitrary scalar field <f>(x) we can by means of a co variant
operation form a vector field with the components d(f>jdx l in the arbitrary
system of coordinates S\ for, from (171), we obtain by differentiation
d
^_
<tf
3x k
^a
dx k dx\
~dx[
ty
lk
(173 )
dx k
dx
where we have used
oL
lfe
(174)
following from (13).
The four-vector
d<f>/dx l is
called the gradient of
grad^
<f>
and
is
written
(175)
g-.
It is analogous to the ordinary gradient vector in three dimensions.
Similarly, from a vector field a t (x) we can form a tensor field of rank
2 with the components da i jdx k in an arbitrary system of coordinates,
for
by
differentiation of (172)
we
get
8x m 8al
da t
(176)
The antisymmetrical combination da k /8xl da l /dx k is also a tensor field
of rank 2 which is called the curl of the vector field a^x). It is denoted by
curl^a
curl
tjfc
{aj
^-^-
177 )
(In the expression curl lA.{aJ we shall of course not sum over i.) The curl
an antisymmetrical tensor. Therefore, in three-dimensional space it
is
corresponds to an axial vector, viz. to curia.
THE THEORY OF RELATIVITY
48
IV,
By
zero.
contraction of the tensor field dajdx k we obtain a tensor of rank
From a vector field a^x) we can thus construct a scalar field
which
is
*$,
8x1
Sx(
(178)
called the divergence of the vector field a {
It is denoted
and
127
by
(179)
analogous to the ordinary three-dimensional divergence div a.
If a l is the gradient of a scalar /f, i.e.
is
al
^A
MQH\
(180)
the divergence of a l becomes
G
Cd*
_*
f
UJ
r_
3x T dx l
dx i
= QA
.
/ 1 f\
i \
(181)
where we have put
I
__ _
__
(182)
The operator
it is
(182), d'Alembert's operator, is thus a covariant operator;
the four-dimensional generalization of Laplace's operator
ai'
In the same way, by means of differentiations we can always form a
n+ 1 from a tensor field of rank w, and by subsequent
tensor field of rank
nl.
As in the special
get a tensor field of rank
of
the
transformation
cases just considered, this is a consequence
equations for tensors together with the equation (173) and the orthogonality
contraction
we then
relations (11)
and
(14).
construct a tensor field
a tensor field of rank
From
a tensor
dt ik /dxl
of rank
1, i.e.
a vector
div 7
which
is
field
3,
ik
of rank 2
field
div.ft-J
ui*/
~,
dXk
called the divergence of the tensor field
Flk
we can thus
and by contraction we obtain
(183)
ik
antisymmetrical, we can obviously form a
completely antisymmetrical tensor field of rank 3
If the tensor field
ourl, w
is
* = curU**}
>
<
184 >
FOUR-DIMENSIONAL FORMULATION OF
128
IV,
48
which is called the curl of the tensor Flk If we interchange any two of
the three independent indices in (184) the expression changes sign.
Thus, curl lkl F is zero if two of the indices are equal and the tensor
.
41
=4
cur\ iki has only
the curl of a vector
independent components.
Fik
is
llc
is
equal to
field, i.e. if
n =
^
the curl of
If
identically zero,
&a
d a ic
eurl l4
/ior\
(185)
=_*-^i,
i.e.
curl lW JF
0.
(186)
way we speak of a pseudo-tensor field when to each point
connected a pseudo-tensor. These are also frequently called
tensor densities, since t lk dx l dx 2 dx^dx^ according to (121), is a tensor
In the same
in space
when
is
a pseudo-tensor. Thus a pseudo-scalar is a tensor density of
rank zero, a pseudo- vector a tensor density of rank 1, etc. To the various
antisymmetrical tensor fields can be attributed dual tensor densities
t
lk
is
which have the same reciprocal connexion as have the tensors and their
dual pseudo-tensors defined in 44. The vector density dual to curl, A
is equal to div F*, where F*k is the tensor density dual to Fik
for,
,
according to (108),
we have
~~
'dx
UJL>
**
2i
r.
lklm
which just means that
49.
~^
fa
\JJL' i.
(/
div,
iV
lnj
F* and
lk
curl lfr/
are dual to each other
Gauss's theorem in four -dimensional space
a(x) is a three-vector field and V a domain in 3-space bounded by
If a
a closed surface
a,
Gauss's theorem in ordinary space
is
expressed by
the equation
[divarfF
a n da,
(188)
where a n is the component of a in the direction of the outward normal
n to the surface element da. (An elementary proof of this theorem is
given in Appendix 1). Gauss's theorem thus permits of transforming
the volume integral o'n the left-hand side of (188) into an integral over
the two-dimensional boundary a of the volume V. If n is a unit vector
in the direction of the outward normal, (188) may also bo written
x
V
where dV
is
given by (107).
(189)
IV,
THE THEORY OF RELATIVITY
49
129
If we choose the surface element in the form of a parallelogram formed
infinitesimal vectors dx L and 8x t lying in the surface a, the surface
by
dx &jc K - rfx K S.r
may be represented by the tensor d<r lK
obtained from (99) by substituting dx and 8x L for a and 6 respectively.
Alternatively the infinitesimal parallelogram may also be represented by
the corresponding axial vector do l defined by (100). Since this vector is
element
perpendicular to the surface element,
'
dv
a d
'
form
89) can also be written in the
'
^xx
8<
(I90)
provided that the sequence of the vectors dx and Sx t is chosen so that the
da lies in the direction of the normal pointing away from
t
axial vector
the domain V.
In this form Gauss's theorem
If a^x)
dimensions.*)*
is
may be immediately generalized to
field and X a domain in (3
a four-vector
four
f-1)-
space bounded by the closed three-dimensional surface V, Gauss's
generalized theorem takes the form
ds
J
whererfS
is
=
J
a*
^\\
dVi
given by (121) and (dx
),
(8x t ),
(ti
and (Aa^-) are three four-\ ectors
boundary space V. The pseudo-four-vector dV is given by
(114) with (a ), (6 t ), (c t ) equal to (dx ), (8^), (Aa; ), respectively.
If a part of the boundary V is a hyperplaneQ defined by .r 4 -- constant,
lying in the
the vectors (dx
Ch se
),
(8^ t ), (Ao: t ) are orthogonal to the
=
=
(fo lf 0,0,0),
A^-
(0,0,^,0).
cte t
8a? i
time axis and we can
(0,rfe 2 ,0,0),
The pseudo-vector (W on the hyperplane O will then have the components
l
dV
(0,0,0,
idx l dx 2 dx^)
(191')
where the plus or the minus sign should be taken according as the
normal to 12, pointing away from the region S, lies in the direction of
the negative or the positive time axis.
If t lK is a space tensor, we have by analogy with
dv
f A.
3595.60
J
a
/w n * da
Sommerfeld, Ann.
/
a
d. Pfnjs.
'* da *
J
a
iat
89)
and
^^
A8
(190)
^'
32, 749 (1910); 33, 649 (1910).
92)
FOUR-DIMEXSIOXAL FORMULATION OF
130
and the analogous equation
f
J
=
^rfS
<>x k
IV,
49
in (3 |-l)-space is
(t lk dVk
[i iK
Z klmn
d^x m ^ n
(193)
The fundamental equations of mechanics for incoherent
matter
As a first application of the mathematical methods developed in the
preceding sections we shall now consider the motion of continuously
50.
distributed matter under the influence of given external forces. In order
to be able to apply to such a system the fundamental equations of motion
of material particles, deduced in Chapter Elf, we shall regard a con-
tinuous mass distribution as a limiting case of a distribution of a very
large number of material particles. If the particles are so small and the
number
of particles per unit volume
so large that our macroscopic
is
measuring instruments cannot distinguish between the single particles,
the mass distribution in an arbitrary system of inertia 8 can be described
by a mass density /^(x, /) which, for practical purposes, may be regarded
as a continuous function of the space and time variables, /^(x, /) is defined
so that /zSF is equal to the total mass inside the \olumo element SI7
at the place x and at the time /. The motion of the matter at any place
and at any time is described by a velocity vector u ^- u(x, f), which is a
x and / Then the mass current density is equal to /zu. In
Newtonian mechanics the mass is a quantity which is conserved. Accord-
function of
is not so in the theory of relativity, when the
acted upon by forces, since the velocity of a given material
particle, and thus its relativistic mass, in this case changes with time. On
the other hand, the proper mass of the matter is conserved in many
ing to formula (III. 22) this
matter
is
when the four-force at any point
the
matter at the place considered,
of
velocity
holds everywhere and at any time.
cases, viz.
The density of proper mass /x
the arbitrary system of inertia
With
Mo
i.e.
is
orthogonal to the fourwhen the equation (57)
i.e.
the proper mass per unit volume, in
according to
is,
^Vd-
(III. 22),
/c
).
connected
(194)
is a given function of x and t.
considering a certain point inside the matter at a given time, we
can introduce the system of inertia 8 relative to which the matter at
/x
like
[A,
Now,
this point is
momentarily at
rest.
/*8
where
all
quantities referring to
In
/S
/*,
(194) reduces to
(195)
are provided with the superscript
0.
THE THEORY OF RELATIVITY
50
IV,
131
In the rest system the proper mass density is thus identical with the
is thus an
mass density. In contradistinction to ^ juJJ
relativistic
invariant. Further,
it is
also easily seen that fJL Q ^J( 1
u 2 /c 2
we
//,
an invariant.
8 has a volume SF,
)
is
consider a small piece of matter which in
system 8 it has a volume SF which, according to
connected with 3V by the equation
For,
if
in the rest
is
SF-
sr
/i
which, by means of
particle expressed in the
^sro-^sp,
--=
two
(19?)
(196), leads to the equation
^/c
or,
(196)
8VJ(I-u*/c*).
The invariant proper mass of the material
systems is then given by
(II. 34),
according to (194), to
/z
2
)
- M
U-
=- invariant,
(198)
== /i.
(199)
Consequently the quantities on the left-hand side of (198) and (199) must
be in valiants.
Let
now
r
nates in A
</>
</>(x, t)
be a given function of the space and time coordi-
We must then distinguish between the local time differentia-
per unit time at a fixed point in
indicating the change of
space and the substantial differential coefficient d(f>/dt indicating the
change per unit time when we follow the matter in its motion.
tion
c>(f)/('f
<f>
We
obviously have
,.,
(200)
In the same way we have for a three-dimensional vector
VV .,
at
field
a(x,
t)
(201)
ct
an equation of the type (200) must hold for each component of a.
now consider the matter which at the time t is situated inside a
closed surface a enclosing a domain with the volume V. At the time
t-\ dt the volume of this material will have been increased by an amount
since
Let us
dV
dt
un
da,
(202)
(7
where the integral on the right-hand side is a surface integral over the
surface cr, and where u n denotes the component of u in the direction of
the outward normal to the surface element da', for every face element da
moves a distance of u dt during the time dt and therefore sweeps through
FOUR-DIMENSIONAL FORMULATION OF
132
By means
a volume u n dtda during this time.
equation (202) can be written
dV
^ J
-'
IV,
50
of Gauss's theorem (188)
udV,
(203)
an equation which must hold
for every part of the material body. Conthe
matter which at the time t lies inside an insidering in particular
finitesimal volume element 8F, we get for the volume dilatation per unit
OV
(204)
ill
Let us begin with the consideration of a system in which the proper
mass is conserved. For any material particle with the volume 87 at the
arbitrary time
we must then have
o
(205)
0,
which, by means of (204), can be written
%+
at
Mo divu-0.
(206)
This equation which expresses the conservation" of the proper mass can
form when we apply equation (200) to the function
also be given another
/x
(x,Z).
We
then obtain
0,
+ divOioU) =
or
ct
As
0.
(207)
the current density of the proper mass, (207) is the continuity
equation expressing that the proper mass in the system considered has
neither sources nor sinks.
/z
is
of (39) we can now define a four-velocity U at any point in
the matter and at any time. As U
i^(x) is a function of the space-
By means
time coordinates, we thus have a four-vector field in (3-f-l)-space. In
the same way, the invariant mass density
defined by (195) and (199)
can be regarded as a scalar field in (3-f 1) -space, H Q being then a function
/u,
of the space-time coordinates in
any system of inertia:
w 2 (ar)/c 2}
= p*(x).
(208)
IV,
THE THEORY OF RELATIVITY
50
By multiplication of the scalar /x/c and the vector
four-vector
if0 rr
c.
133
U we obtain a new
T
(209)
^f,
which may be called the four-current density of proper mass. According
and (198) we get for the components of c i
to (39)
(210)
and the continuity equation
(207) can be written in the tensor
form
^i^i^ffi^O.
c
ox
dx t
(211)
The left-hand side of (21 1)
is equal to the four-dimensional divergence
four-current
the
of
density, and the covariance of this equation
(178)
under rotations in (3+l)-space is therefore evident.
The
forces acting
on the different parts of continuously distributed
matter are partly external impressed forces, partly elastic forces acting
between neighbouring parts in the continuum. In this chapter, we shall
and postpone the conHere we therefore treat
completely neglect the last-mentioned forces
sideration of the elastic forces to Chapter VI.
the matter as a kind of incoherent dust. The impressed forces, however,
are assumed to be volume forces which in every system of inertia S can
be described by a force density f so defined that f 8 V is equal to the force
on the volume element 8F.
Let us now consider the motion of a small material particle with the
volume 8V arid with the proper mass Q 8V =
SF. When U is the
JJL
four-velocity, the
four-momentum of
Pl
and
for the four-force (54)
Since
8F
= ^o 8FC7
-f7r^rx\
III
s*\
an invariant and 2^
is
is,
according to
(50),
(212)
we obtain by means of (196)
*r*>
& si\
I
/LL
this particle
ni
is
(
I
>
^^) 8F
st
<^
213 )
'
a four-vector, the quantity
form a four- vector, the four-force density. The spatial components of/ are equal to the ordinary force density and/4 is equal to the
mechanical work done per unit of time and of volume multiplied by ijc.
will also
FOUR-DIMENSIONAL FORMULATION OF
134
The motion of the
(55).
is
particle considered is
Substituting from
conserved,
i.e.
when -
(p?
SF)
is
given by
(2 1 4).
described
by equation
the proper mass
when
0,
P-^=f
where f
now
(212), (213) in (55) gives,
50
IV,
(215)
Since /x and dr are invariants, both sides of this
equation represent four-vectors. The first three equations (215) are
again the equations of motion, while the fourth equation expresses the
theorem of conservation of energy. According to (214) and
(39)
which, as discussed in 38, is essential for the conservation of the proper
mass. By multiplying (215) by U and summing over i, the left-hand
side becomes zero on account of (41'), and the equations (215) there1/
fore appear to be compatible with (216).
The fundamental equations of mechanics assume the simple form
(215) only in systems in which the proper mass is conserved. If we drop
this assumption we have, in accordance with (59), to use the following
expression for the four-force density:
(217)
where q is the amount of non -mechanical energy developed per unit of
volume and time, so that (f.u)-f- q represents the total effect per unit
volume. In this case the left-hand side of (55) becomes
.
dr
dr
ar
(218)
the volume of the particle and dr is the time increase, both
measured in the rest system *S we obtain by applying equation (204) in
Since
8F
is
the rest system
= Srdivu
dr
(219)
where div means differentiations with respect to the space coordinates
Q
in /S. Tn the rest system u = 0, on the other hand, the derivatives
(X
of u with respect to the coordinates need not be zero. By means of the
expresvsions (39) for U and t/J we now get, since the quantity dUJdx is
L )
invariant,
^n = ^ = divu.
~
~r 7 o
dx k
'dx\
(220)
THE THEORY OF RELATIVITY
50
IV,
Using the equations (217)-(220) in
(55)
135
we then obtain
in the general
case, instead of (215),
=L
where /
now given by
is
(221)
Since, furthermore,
(217).
dr
i'x^
(221) can also be written
JL^of/.^)
(223)
_._f^
which thus represent the fundamental equations governing the motion
of a continuous mass distribution in the general case in which the proper
mass
may
From
where
not be conserved.
(217)
of course
is
is
and
(39)
we now obtain
the non-mechanical effect in the rest system. This quantity
an invariant, and we ^et from (224)
q-M(l-u*lc*)
(225)
Multiplying this equation by the invariant
8
TAJ
- SFAr
we again have
the equation (66), since &Q ~ q SFAf is the amount of
heat conveyed to a certain material particle during the time A. If we
multiply (223) by U^ and sum,
or,
we
obtain, using (41), (41'),
and
(224),
according to (209) and (210),
These equations are thus generalizations of the equations (211) and (207)
where the system contains sources for the proper mass, and
for the case
the expression <//c 2 tor the source density
general theorem (III. 74).
In the preceding considerations
it
is
in
accordance with Einstein 's
has been tacitly assumed that no
heat conduction takes place in the matter, so that the transport of heat
therefore occurs only by means of convection. Fn the more general case
where heat conduction
also occurs, it
must be taken
into account that
FOUR-DIMENSIONAL FORMULATION OF
136
IV,
50
the energy transported by means of heat conduction represents an extra,
non-material momentum which would cause a change in the first three
equations (223).
51.
The
kinetic
energy-momentum tensor
The quantity appearing on the left-hand
= vU Vk
e ik
is
side of (223)
(228)
a symmetrical tensor of rank 2 which is called the kinetic energytensor. f (223) can thus be written
momentum
=
f^
cx k
i
e.
(229)
the four-force density is equal to the divergence of the kinetic energytensor. By means of (39), (198), and (199) we obtain the
momentum
following expressions for the components of this tensor:
-^ = -h,
(230)
where h is the energy density or, more precisely, the sum of the kinetic
energy and the proper energy per unit volume. Multiplication of (230)
by 8V makes the right-hand side, apart from the minus sign, equal to
the energy of the small material particle inside the volume 8V.
The three components 64l form the components of a spatial vector
which can be written
(231)
The three components
0, 4
can in the same
way
be written
(^^^) = -^^ r -g.
hi view of the equations (229)
**
,0 4l
must be
i
7m
(236)
and
and
OJic
(238)) the quantities
interj)reted as energy current density
espectively.
involves
(cf.
t4
(232)
g
and momentum
density,
The symmetry of the energy-momentum tensor which
4t
or
=-u
c2
(233)
then simply expresses that the energy h corresponds to a mass h/c 2
f
H. Mmkowski, GotL Nachr.,
p.
53 (1908); Math. Ann. 68, 472 (1910).
THE THEORY OF RELATIVITY
51
IV,
The
spatial part 6 iK of the
Q
"<*
137
energy-momentum tensor can be written
U K __
p, G U
f9<U\
jir^ict)
(234)
where g and U K are the components of the space vectors g and u.
L
Just as the quantities (231) represent the energy current density, each
row of the space tensor IK in (234), e.g.
(0*AM = giV>
(23- >)
can be interpreted as the current density of the momentum component y
Therefore IK is also called the momentum current tensor.
The fourth equation (229), expressing the energy conservation, can
now, by means of (230), (231), and (217), be written
+clivOzu)
(237)
This equation is exactly analogous to equation (227). While q/c* expresses the source density of proper mass, we see that the source density
for relativistic
mass
is
((f .u)+^)/c
2
.
accordance with
this is in
Again
Einstein's relation (III. 74).
In the same way, the first three equations (229), the equations of
motion, can be written
%4.^~
(238)
fl+c^-'*'
^+
div(i/ 4
where we have made use of (217),
u)=/
and
(232),
(239)
(234).
Equation (239) plays
momentum as does the analogous equation (23(>) for
regulates the flow of momentum in the material con-
the same role for the
the energy. It
tinuum, the force density
appearing here as a source of momentum.
the corresponding equation
we multiply (238) by X K and subtract
obtained by interchanging and
we get
If
r\
^t
On
/c,
r\
(9^ K -g K x
i)
+ -^~(0
iX
x K -0 KX x )-0 lK +0 KI>
l
- f x K ~fK x
L
(240)
account of the symmetry of the tensor d LK the last two terms on the
,
left-hand side cancel each other.
Introducing the angular
momentum
density and the density of momentum of force by
>IK
^if/K-^f/p
(241)
FOUR-DIMENSIONAL FORMULATION
138
(cf.
IV,
51
equations (82)-(86)), (240) can be written
w+?*^ =
where we have applied the expression (234)
also be given the form
^+(u.grad)m w +m
and,
when we multiply
(244)
l(t
(243)
>
for
divu
LK
This equation can
d iK
(244)
by BV, we obtain, by means of
(200)
and
(204),
*
(n, lK
t
W)^d
lK
?>V.
(245)
This equation expresses the angular momentum theorem for a small
material particle with the volume 8V. We thus have seen that the
symmetry of the kinetic energy-momentum tensor is a very important
property, since the symmetry of the spatial part is essential for the
validity of the angular
momentum theorem
-~
equation t4
4t i e. (233),
the inertia of energy.
,
The
is
in its usual form, while the
an expression
for Einstein's
theorem of
kinetic energy -momentum tensor satisfies the relation
o*
where h
is
uk -
uk uk
=.
-pw\ =, _ w
the energy density in the rest system.
/t)
(246)
ELECTRODYNAMICS IN THE VACUUM
The fundamental equations Of electrodynamics in the
vacuum. Four -current density for electric charge
IN Chapter III we have seen that it is necessary, to change the funda-
52.
mental equations of mechanics in order to bring them into accordance
with the principle of relativity. This is not so with the equations of
electrodynamics in the vacuum, the Maxwell equations, which, as we
shall see, are already co variant under Lorentz transformations. f
Let us imagine two teams of experimental physicists who have installed their laboratories in
two
different inertial systems
S and
8'
and
who independently
means
are performing electromagnetic experiments. By
of electrically charged test bodies and magnetic compass needles
the physicists in S will be able in the customary way to determine the
electric field vector E and the magnetic field vector
as a function of
the space-time coordinates x and t in S. By the same procedure the
physicists in 8' will be able to determine electric and magnetic field
and H' as functions of the coordinates x' and V in S'. Furthermore, the two groups of physicists can, independently of each other,
determine the charge densities p and p in S and S' In the present
chapter we shall consider only electromagnetic phenomena in the vacuum
as neither conductors nor dielectric and magnetic substances are present
here, the only type of electric currents occurring are convection currents.
vectors E'
8 and
pu and p'u', where u and
u' are the velocities with which the charges move in 8 and S', respectively. All these quantities will be certain functions of the space and
time coordinates in 8 and 8'.
Now, according to the principle of relativity, the equations deterThe current
densities in
8' will thus be
mining the fields as functions of the charge distribution should have the
same form in 8 and 8' ( Consequently, both groups of physicists should as
a result of their experiments be led to the Maxweli-Lorentz field equations for empty space. In S we thus have, applying Heaviside's units,
.
,^TT
-~^0,
c dt
(la)
i^ = ^,
(16)
c dt
t H. Fomraip, d.It. 140, 1504 (1905), Rend. Pal. 21, 129 (1906), A. Einstein, Ann.
Fhys. 17, 891 (1905), H. Mmkowski, see ref., Chap. IV, p. 136.
d.
ELECTRODYNAMICS IN VACUUM
140
and
in S'
we
V,
52
get equations resulting from (1) by adding a prime to all
(1) are identical with the fundamental equa-
The equations
quantities.
tions of Lorentz's classical electron theory.
While the connexion between u and u' is given by (II. 55),
and
yet know the connexion between p and />' or between
we do not
H, on the
one hand, and E' and H' on the other. However, it is one of the most
fundamental experiences that electric charge is conserved, a property
which, in analogy to (IV. 207), can be expressed by the continuity
~
equation
P u)
0.
(2)
is a simple consequence of Maxwell's equations
a
similar
equation must be valid in S viz.
Obviously,
This equation
(J6).
+ divVii') =
0.
(2')
The connexion between p and p must now be such that
charge and current distributions (2') is a consequence of
four quantities in S by
for arbitrary
(2).
Defining
(3)
and analogous quantities
s[
in S', (2)
and
(2')
be written
may
~ =
and
shown that,
all possible charge and current
s and s[ must be given by
In Appendix 2
it is
(4)
0.
if (4') is
(4')
to be a consequence of (4) for
distributions, the connexion
between
where
are the coefficients in the coordinate transformation (IV. 3)
connecting the systems S and S'. Thus, s i is a four-vector, called the
oc
lk
four-current density, and (4) expresses that the divergence of the fourcurrent density is zero (cf. IV. 178).
/
If
we multiply
the invariant
si sl
by
1,
we
obtain with the help of
=
(3)
$Xi
(6)
an invariant
- P 02
(?)
ELECTRODYNAMICS IN VACUUM
52
V,
where p
is
By means
$. Hence we
the charge density in the rest system
of
(8),
the equation
141
get
can be written in the form
(3)
,=^,
U
the four-velocity defined by (IV. 39). Equation (9) is quite
where l
analogous to the expression (IV. 209) for the four-current density of the
proper mass.
is
Let us now consider the charge p 8V connected with a material volume
element 8V. If SF denotes the corresponding volume in the rest system,
2
2
we have 8F
SF\/(1 u /c ), which together with (8) gives
pSF-/>3F.
(10)
Hence, the electric charge of a certain material volume element is an
invariant, and the same is therefore also true for the total charge of a
material body. This important theorem of the invariance of electric
charge is thus a consequence of the validity of the continuity equation
system of inertia. It can also be made plausible by the following
reasoning. Consider a charged particle of charge e originally at rest in S.
Under the action of a force the particle is accelerated until it has the same
velocity v as has S' relative to S. Since the charge of a particle is conin every
served during acceleration, the particle has still the charge e relative
to S. On the other hand, the particle now has the velocity zero
relative to S' and, since
it is
now
in the
same
situation relative to
previously was relative to S, the charge e' of the particle relative to S' must be assumed to be equal to the constant charge e relative
as
it
to S.
Therefore
we must have
at
any time
e'
e,
in
agreement with
the equation (10).
Covariance of the fundamental equations of electrodynamics
under Lorentz transformations. The electromagnetic field
53.
tensor
In every system of inertia S we now define a quantity
Flk
by the
equations
F = -Fkl
ik
(Fm
Fn F12 =
)
i.e.
H.
Hv
(Fn
-Hu
-iEx
-Hx
*
Tji
l&y
F Fn =
H,
/*
1>&Z
E,
(11)
ELECTRODYNAMICS IN VACUUM
142
The equations
(1 a)
53
V,
can then be written
CXt
O^r
O^k,
Since the expression on the left-hand side in (13) is completely antisymmetric in the three indices i, k, /, (13) represents only four indepen-
dent equations which are obtained, for example, by putting (i, k, I) equal
to (1,2,3), (4,2,3), (4,3,1), (4,1,2), respectively. It is easily verified
that these four equations are identical with the four equations (I a).
From the general validity of the equations (4) in every system of
we concluded that the quantities s are the components of a fourvector. In the same way we can conclude that the quantities Ftk must
inertia
transform as the components of an antisymmetrical tensor if the equations (13), as should be required, are valid in every system of inertia.
The tensor Flk thus defined is
called the electromagnetic field tensor, and
or
the equations (13)
(la) express that the curl of this tensor is zero
(cf. IV. 184).
On
account of (IV. 187) the equations (13)
r JP*
v*
div
t
where F*L
->
H,
is
H -*
^=
'*
may
also be written
n
(14)
0,
the pseudo-tensor dual to Flk obtained by the substitution
E in the expression (12) for lk
(11) between Flk and E H is the same as in (IV. 80, 80' ),
which indicates that E and H behave as a polar and an axial vector,
respectively, under pure spatial transformations. For a general Lorentz
transformation without rotation we have the transformation equations
(IV. 81') for E and H. These equations may also be written in the form
The connexion
(15)
H'
The
division of the field into
an
electric
and a magnetic
field,
which
is
forced upon us by our measuring instruments, thus has no absolute
If, for example, we have a purely electrostatic field in S, i.e.
there will, according to (15), be a magnetic field H' =/in S'.
meaning.
H=
0,
ELECTRODYNAMICS IN VACUUM
53
V,
This
is
also quite clear
from a physical point of view, since a purely
means that
electrostatic field in S'
143
all
charges are at rest relative to S.
v.
Relative to S' the charges will therefore move with the velocity
Consequently we have in 8' a stationary current which causes a magnetic
field in S'.
By means
of
(3)
and
(12) the
can now be written
second set of Maxwell's equations
(1 6)
Since the left-hand side in (16) is the divergence of the electromagnetic
tensor (cf. IV. 183), the left-hand and the right-hand sides are- transformed in the same way, viz. as a four- vector. Hence, the co variance
field
of the equations
(1 b) is
a consequence of the co variance of equations (la)
and of the continuity equation (4). This is a strong argument in favour of
the exact validity of (16) and it is seen, in particular, that the term
1
dE
9
~c~dt
Maxwell's displacement current,
is absolutely necessary for the covariance of the equations (1ft).
If we form the di\ ergence of the vector equation ( 1 6) we get, on account
of the antisymmetry of the electromagnetic field tensor,
-^0,
(17)
'dx l <dx k
i
e.
54.
the continuity equation
(4).
The four -potential. Gauge transformation
As is well known, E and
be written
in
H can, as a consequence of the equations (1 a),
the form
H-
curl
way
grad<A
(18)
tit
and the scalar potential can always be
that they satisfy the Lorentz condition
where the vector potential
chosen in such a
E =
A,
</>
divA + ~^-:0.
C ut
This can be accomplished in every system of inertia. If we
A l in every system of inertia by
(19)
now define
four quantities
A-MA,^),
(20)
ELECTRODYNAMICS IN VACUUM
144
and
(18)
(19)
can be written
= 0.
'
(22)
dx l
Since
54
V,
Flk is a tensor, it must be possible to choose the potentials in the
different systems of inertia such that the
are transformed as the
components of a four-vector, the four-potential. According to
(21)
and
(22) the electromagnetic field tensor is equal to the curl of the fourpotential which has a divergence equal to zero. Moreover, from (IV. 184186) it follows that Maxwell's equations (13) are a consequence of (21).
When Flk is given, the four-potential A is by no means uniquely deter(21); for, if A satisfies (21), the functions
L
mined by
{23)
*-*'+%;
where
is
an arbitrary
The transformation
scalar, will also satisfy (21).
a gauge transformation and the
measurable quantities Flk are invariant under such transformations.
The Lorentz condition restricts the class of permitted gauge transformations, but there is still a great variety of potentials A satisfying this
(23) is called
condition.
Substitution from (23) in (22) leads to the condition
=
If
if
is
D<A
r
an arbitrary solution of (24), A*
satisfies
0.
(24)'
thus satisfy both (21) and (22)
will
these equations.
Inserting (21) in (16)
we
get, using (22),
dx k fa k
=*,
D^t = ~ s
or
(25);
l'
(26)
Any solution of (26) which simultaneously satisfies (22) gives,
of (21), a solution of Maxwell's equations (13) and (16).
55.
Four-dimensional
representation
integral
of
by means
the
four-
potential
The equations
(3-}-l)-space;
have the form of usual potential equations in
the solutions can therefore be found by a method exactly
(25)
analogous to that applied in three dimensions. f
f A, Sommerfeld,
Ann.
d.
We
Phys. 33, 649 (1910).
shall first write
ELECTRODYNAMICS IN VACUUM
55
V,
145
down
the solution of the equations (25) on the assumption that all four
coordinates x t are real, so that the four-dimensional space is Euclidean.
Let
*,
= *.-*.(/)
(27)
be a four-vector connecting a fixed point P with coordinates x^P) and a
If R 2 = (R R
denotes the
variable point with coordinates x %
of
shows
the
distance
between
these
a
calculation
points
simple
square
.
that
^=-
^QA-0n< =
Now
us put
let
t/r
02
x z (P). Further, if *jj(x) and
= (#J, we have
regular functions of (x)
any point x l
at
<f>(x)
are
two arbitrary
dx
in this equation and integrate over the whole
lies outside a (three-dimensional) sphere
1/jR
four-dimensional space which
R%
= R-R
with radius a and with the point
use
R = x^x^P)
i
on account of
x (P})(x
(x
P as centre.
x (P))
(29)
cfi
If in the right-hand side
as integration variables instead of # t
we thus
we
get,
(28),
x l dx 2 dx 3
(30)
where the integration has to be extended over the four-dimensional region
for
which
Rt^K^^a*.
(31)
Since the integrand on the right-hand side of (30) is a sum of partial
differential coefficients, the integral can be transformed into a three-
dimensional integral over the sphere (29), provided that the function
vanishes sufficiently rapidly at infinity. For the first term on the righthand side of (30) we then obtain
</>
(32)
where the integration
ingly,
)+
and
)-
is
to be extended over the sphere (29). Accordin the function in the brackets is to
means that
be put equal to the values
3595.60
ELECTRODYNAMICS IN VACUUM
146
and the domain of integration over the variables R 2
the inequality
^ ^^<
V,
72 3
55
# 4 is defined by
fl2 _
(34)
We get expressions similar to (32) for the three other terms corresponding
4 on the right-hand side of (30); they are obtained from (32)
by cyclic permutation of the indices (1,2, 3, 4).
(The transformations just performed in (30) correspond to Green's
to k
2, 3,
theorem in three dimensions
Subsequently, letting a -^
domain of integration in (32)
the volume of the three-dimensional
0,
\\ill
1
tend to zero as a 3 and, since
in the brackets
)*,
"
"
a2
R-
the second term inside the brackets will tend to zero.
~
Since, moreover,
"
--
r>
(35}
I''
'
^Z A 5* -"**"'
we obtain
for (32) in the limits ot very small
dR
!zPi)
using (33),
dR.A dlt 4
(36)
at the point P. The domain of
1
of
the
the
interior
sphere p\ ~-^ a' The three other
integration (34)
terms obtained by cyclic permutation of the indices 1 2, 3, 4 are obviously
equal to the first term. Therefore, introducing polar coordinates in the
where
(f)(P) is
the value of the function
<j>
is
integration,
we
finally obtain for the right-hand side of (30)
a
477
~T
,,
v( a ~~~Pi)Pi "Pi
"
(37)
and
(30)
becomes
47r </(P)
-L
(38)
where d*x
is an abbreviation of dx l dx 2 dx^dx^. This equation holds for
If in particular
is the function A satisfying
any regular function
</>.
(26),
we thus
</>
get the formula
"
4
5-,d *,
whicli allows us to calculate
A at an arbitrary point P
known at every point.
l
dimensional space when s is
Until now we have assumed the variables
t
x4
.r
in the four-
to be real.
However,
problems the four-current density s is not given for
but only for purely imaginary .r 4 values corresponding to t
t(P).
in actual physical
real
(39)
<
ELECTRODYNAMICS IN VACUUM
55
V,
In the complex # 4 -plane
s t is
147
thus given only in the fat-faced part of the
therefore deform the original path of
We
imaginary axis in Fig. 13.
L around
this part of the
s t in the
function
imaginary axis, using the analytic continuation of the
integrand of (39). The expression (39) will then still be a solution of (26).
integration along the real axis into a loop
plane
ict(P)
-100
FIG. 13.
Since
we
(40)
get for the divergence of
=~
4772
~df(p)
A
Sl
(P) in (39)
d *x
'
a^T"
J fal
5*
(41)
by partial integration and application of (4). The solution (39) thus
also satisfies the Lorentz condition (22).
From (21), (39), (40), and (35) we thus obtain for the electromagnetic
field
tensor
2ntfu (P)
P<
r,
* Si
d*X.
(42)
Retarded potentials. Lienard-Wiechert's potentials for point
charges
56.
In performing the integrations in (39) the sequence of the integrations
may be chosen arbitrarily and we shall first perform the integration
over the ^-variable along the path L in Fig. 13, keeping (x v % 2 x3 )
,
ELECTRODYNAMICS IN VACUUM
148
constant.
If r
space points
56
V,
x(P)j denotes the spatial distance between the
to the event points (x ) and
|x
x and x(P) corresponding
we have
(43)
(Xt-Xt(P)+ir)(Xt-Xt(P)-ir).
Therefore the quantity 1/B
loop L, viz. the point
as a function of # 4 has a pole inside the
=x
^_
ir
in the complex # 4 -plane. Since the integrand in (39) has no other poles
inside L, the path of integration can be deformed into a contour around
we then obtain
the point (44) and, by means of Cauchy's theorem,
x t(*)
Consequently the potentials
(39)
x\
**
xi-=x t (P)-ir
assume the form
-^ dV,
(46)
where the integration has to be extended over the usual three-dimenThe function s l in the integrand is not to be taken at the time
at
the
time t(P)r/c, corresponding to the fact that all electrobut
t(P)>
magnetic disturbances are propagated with the finite velocity c. There-
sional space.
fore the potentials (46) are called retarded potentials. If in (39) and (45)
we had integrated along a curve obtained from L by reflection with
ict(P), we would have obtained another solution
respect to the point # 4
of (26) corresponding to advanced potentials. This solution, which con-
nects the field at a certain point at a certain time with the future charge
and current distributions, has, however, usually no immediate physical
application.
Let us
now
consider the potential of a point charge e in arbitrary
of the point charge being given as functions
t
motion, the coordinates
of the time *:
t (<).
The
(47)
four potentials can then be obtained from (46), but, since s l in (46)
on account of the retarded time variable has a rather complicated
dependence on the variables of integration, it is easier in this case to go
back to equation (39). Here we can now first perform the integration
over the space coordinates and, since s is zero everywhere but at the
points satisfying (47), we have, according to (3) and (IV. 39),
t
f Si
J
dx dx dx
*
eU
ELECTRODYNAMICS IN VACUUM
56
V,
Here u
the velocity of the point charge given by
is
(dxjdt)
149
(47),
is
the corresponding four-velocity, and
now functions of t or of the purely imaginary
but by analytical continuation they can also be defined
'outside the imaginary axis in the # 4 -plane. From (39) we thus get
All these quantities are
variable # 4
The integrand again has a pole at the point (44) in the complex plane,
and near this # 4 -value the denominator has the form
R2
_d]P (x 4 _.
~~
dx^
where the functions
=
o K>
p
dHjf
k
0/^4
a# 4
<y~P
TJ
^{Ifr Ujc
~T
ao; 4
9 P TT V'*
^*lk "k
/KA\
'
\*^^/
tc
and
r(^) should be taken for the value of # 4 for which x^x^(P)-\-ir
By means of Cauchy's theorem we thus obtain|
Here (B k ) denotes the four- vector leading from the fixed event point
to that point Q on the time track
*
0.
(52)
*i(r)
of the point charge in which the retrograde light cone
B =B B =
2
originating from
(53)
P intersects the time track.
Also
should be taken at
the point Q.
If r denotes the spatial part of the four-vector (JBJ leading from
Q we get for the denominator in (51)
-
77
by means of
477A(P)
'""VU-^/c
(IV. 39)
t-i
and
2
)
(44).
<J(l-u*/c*)
The equations
(u.r)+rc
^/(i-u
(51)
/c
r+(u.r)/c
r+(u.r)lc
Mmkowski,
can thus be written
eu/c
f H.
P to
see ref., Chap. IV, p. 136.
(55)
ELECTRODYNAMICS IN VACUUM
150
The equations
(55) represent
56
V,
Lienard-Wiechert's potentials of a moving
point charge.
If
we wish
to calculate the electromagnetic field tensor lk (P) at the
by means of (51) we must keep in mind that the proper
event point P
time r corresponding to the point Q is a function of the coordinates of the
and (52), by
point P defined by the equation (53) or, according to (27)
(xM-xAPMxM-xAP)) By
0.
(56)
we obtain
differentiation of this equation with respect to x k (P)
Kk
Sr
/K -v
(57)
From
(21)
and
(51)
toF
we then
get
(P)' -*-!*-'*
(
<h\U,R
using (IV. 41),
or,
dU
dl*
'
Equation
(58) could also
u'-*^'
be obtained directly from (42)
if
we
(B8 >
first inte-
grate over the space coordinates and subsequently use Cauchy 's theorem
in the following integration along the curve L in the # 4 -plane. Here it
should be remembered, however, that the function
second order at the point (44).
l/7i
has a pole of
Since F, k according to (58) has the form
^R^-E^,
where a
is
a four-vector,
we have, according
(59)
to (IV. 110, 111),
^^=0,
F
where F*k
(60)
is the pseudo-tensor dual to
lk which is obtained from (12) by
the substitution (H, E) -> ( E, H). The equation (60) is thus identical
with the equation
(E.H)
The
all
electric
and magnetic
0.
field vectors are
(61)
thus everywhere and in
systems of inertia perpendicular to each other. Further,
we
find
by
ELECTRODYNAMICS IN VACUUM
56
V,
means of a simple
calculation, applying (12), (58),
and
151
(53),
0/>2 r 4
(
62 )
2
2
is always negative for the field
|E
Consequently the invariant |H
of an arbitrarily moving point charge. This, together with (61), involves
that for an arbitrary event point P we can always choose such a system of
|
inertia that the
components of the magnetic
and
vector in this system
one needs only
field
are zero at the point P. In order to obtain H'
to choose
TT\
r ,p
O'in (15)
(63)
-p
this is physically possible, since
on account of (61) and
_-=
= cEH
ell
<
(64)
(62).
The field of a uniformly moving point charge
Let us now in particular consider the field of a point charge moving
with constant velocity. The time track of the charge is here a straight
line in (3+ l)-space with a direction defined by the constant four-velocity
57.
Since dl\ldr
l^.
---
0, (58)
4nF. k
reduces to
CC
-
(l^R^
Rk U
(RJL
).
(65)
a suitable choice of the origin in the system of coordinates S we can
always ensure that the # 4 -axis and the time track are lying in the same
plane In Fig. 14, which gives a representation of this plane, the line L
By
represents the time track of the particle, and Q is the point at which the
intersects the time
retrograde light cone from the arbitrary point
the vector leading from P to Q. If A is the projection
on L of the point P the vector leading from A to P, with components
l)
l)
is orthogonal to the four-velocity l\, and x(
is the projection of the
track.
Hence
is
vector
Therefore
It is
(66).
on a direction perpendicular to L\ hence
we have
( *>
-^
U RJ.
immediately seen that the equation
From
(67)
and
(53)
we
(67) satisfies
further obtain
(67)
both equations
ELECTRODYNAMICS IN VACUUM
152
V,
57
In the rest system S' of the point charge the ^-axis is parallel to fy and
to the time track L. In this system we therefore have
^'-(r',0),
where r
point
get
/>'
(68)
the space vector connecting the point charge with the space
corresponding to the event P. Since Ut R l is an invariant we
is
by means of
(IV. 39), (27),
and
(44),
cr'
(r'
(69)
|r'|),
FIG. 14.
in
accordance with
(67')
R,
and
(68).
Moreover, we get from (67)
Uk -R k U = -( x
t
so that (65) can be written
(70)
^
replace the vector x
by a vector
on
the
to
from
an
curve
L
the
arbitrary point
point P, since all
leading
such vectors have the form
In this expression
we can obviously
d*>+aU,,
(71)
a constant. In an arbitrary system of coordinates S we now
such that the time component # 42) is zero. In Fig. 14
this vector is represented by the line BP. In $ we thus have
where a
is
choose a vector x
,(2)
x\
r,0),
(72)
V,
ELECTRODYNAMICS IN VACUUM
57
153
the space vector leading from the simultaneous position of the
point charge to the point of observation p corresponding to the event P.
(The r used here thus differs from the r used in the Li6nard-Wiechert
where r
is
potentials which
was the
spatial part of the vector
have
E^)
We
=-(U-lW),
which by means of
and (IV. 39) leads to-the following expresand magnetic field vectors:
er
tr
A
'
Fig. 14
(73)
(12), (72),
sions for the electric
From
therefore
we
uxr
tiA\
'
two vectors x(2) and x(l} have the same
The connexion between r' and r is therefore
see that the
space components r' in S'.
obtained simply by using the reciprocal of the transformation equation
(IV. 29) for the vector with components
Since the velocity of S' relative to
r'
:=
I
r
I
hence
U r )(l
parallel to u, respectively, (75)
(r'
get
2
/ 02 ))
+ iHlE^U.
r'
into
nK\
tJJ
(76)
components perpendicular and
-^'
/J
Hence the connexion between r and
can be written
traction in the direction
we
V^""^
Decomposing the vectors r and
The
u,
r'
'*
is
r'
(77)
corresponds to a Lorentz con-
(cf. II. 35).
electric field vector lies in the direction of the radius vector r,
is perpendicular both to r and u. The equations (74) are easily
seen to be in agreement with the equations (61) and (62). The surfaces
2
with constant values of the quantity (E 2
are obviously rotation
)
while
~H
ellipsoids,
Heaviside ellipsoids, which are obtained from
constant by a Lorentz contraction in the direction of motion
so-called
spheres r'
of the point charge.
The expressions (74) may also be obtained in a much simpler way by
means of the transformation equations (IV. 81') for the electromagnetic
field vectors.
If S' denotes the rest
system of the point charge, the
ELECTRODYNAMICS IN VACUUM
154
electric field is spherically
symmetrical and the magnetic
V,
field is
'
S', i.e.
57
zero in
4nE'
Hence, from (IV.
81')
and
= er
~,
H'
0.
(78)
(78),
(UXr/)
Since the equations reciprocal to (75) are
we again obtain the expression (74) for the electromagnetic
of a point charge in uniform motion.
From the latter deduction it results that the equations (74) are valid
also everywhere outside a charged sphere provided that the charge
(cf. IT. 35'),
field
distribution in the rest system
is spherically symmetrical
r denotes the distance from the centre of the sphere while
charge on the sphere.
In the same way we can
accelerated' point charge,
motion
(cf.
29).
En this case
e is
the total
also determine the field of a 'uniformly
i.e.
a point charge performing a hyperbolic
either we can
Again we can use two different methods-
directly apply the equation (58) which is valid for an arbitrarily moving
charge, or we can introduce a system of coordinates 8* which follows the
charge in its motion and solve Maxwell's equations in this system, and
then transform back to the system S. However, the system of coordinates $* will not be a system of inertia and, consequently, the
application of this latter method requires the development of a general
theory of relativity which allows the use of arbitrarily moving systems
of coordinates. The necessary tools for an application of the second
method
will
be provided in
97 and 115.
The electromagnetic forces acting on charged matter
By means of the equations deduced in the preceding sections we
58.
distribution.
Now we
of a given field
are
by an
arbitrary charge and current
shall consider the opposite problem of the influence
able to calculate the field created
Flk on the motion of electrically charged matter.
Our first
V,
ELECTRODYNAMICS IN VACUUM
58
155
task will be to determine the force on an electrically charged particle
of charge z moving in a given electromagnetic field with the velocity u
system of inertia S. In accordance with the method
shall now introduce the inertial system 8 in which
outlined in 29
the particle is momentarily at rest. In this system the force F must be
relative to a certain
we
F
by the very
==
eE,
(79)
definition of the electric field vector
in this system.
Introducing the equation (79) into (III. 43), and keeping in mind that
the velocity of >S' relative to S js equal to the particle velocity u, one
easily finds
magnetic
by means of the transformation equations
field vectors,
(15) for the electro-
Lorentz's expression for the force
S, viz.
r
e
F in the system
-,
E + -(uxH)L
(80)
This expression thus follows without extra hypotheses from the principle
of relativity.
This deduction becomes
representation and try
much
if
simpler
we
four-force. In the case considered, the proper
four-force
is
^o
(F,0).
the four-velocity of the particle
is
mass
thus defined by (IV. 54); therefore
system
If
use the four-dimensional
to determine the expression for Minkowski's
is
conserved and the
we have
in the rest
(81)
we can now form a four-vector
with the components
*FikUk
(82)
In the rest system & we have
l/;
(0,0,0,fc),
and the components of the four-vector
to (12),
(79), (84),
and
8 thus become, according
e
c
From
(82) in
(83)
and
(81)
we
ik
see that the
components of the four-vectors
but two four-vectors whose
(82) are equal in the rest system,
components are equal
identical.
Therefore
in
one system of coordinates are altogether
we must have
F = -Fik
k
lk U,.
t
i
in every
system of coordinates.
(85))
\
ELECTRODYNAMICS IN VACUUM
156
Since
Fki we
Flk
V,
58
have
F U = e-F UtU =
t
(86)
ilc
L>
in agreement with (IV. 57), and the equations of motion of the particle
are given by (IV. 56) and (85)
,
ar
If
we
calculate the
(IV. 39),
we
equation
(80).
components of
^ IK ^ K
v*'/
in (85)
get the equations (IV. 54) with
by means of (12) and
given by the Lorentz
Now
consider a continuous distribution of charged matter, with a
four-current density (3), (9), i.e.
_orr
(88)
in a given external field. The problem is then to find the expression for
the four-force density fl defined by (IV. 214). This question can be
solved on similar lines as above.
Consider a definite point in space at a definite time; the charged
matter at this point is moving with a certain velocity. Now, let 8 Q be
the
momentary
rest
$. These components
(IV. 2-14)
(0, 0, 0,
F k sk = (pE, 0)
has the components
in
The comand
the fourip)
system of the matter at this point.
ponents of s i in this system are then s
vector
(90)
are equal to the corresponding components
of the four-force in
SQ
/?
for the force-density in the rest
(,0);
(91)
system must be given by
fo
^pE
o
(92)
by the definition of the electric field vector.
The four-force density must therefore be equal to the four- vector
(89) in
every system of coordinates, hence
/*
From
(88), and
(IV. 214), (93),
sion for the force density
***>'
(12) we immediately get Lorentz's
(93)
expres-
(94)
ELECTRODYNAMICS IN VACUUM
58
V,
Further,
we
view of the antisymmetry of the
get, in
/,
U = Ftk sk U =
t
U Flk Uk =
t
field
157
tensor
(95)
0,
which, as discussed in 50, means that the proper mass is conserved in
this case. Therefore the equations of motion have the simple form
j TT
(IV. 215).
59. Variational principle of
electrodynamics
and
the equations of motion (96) may be
equations
derived from a certain variational principle formulated by Weyl and
Born.t If we put'
_ aA* 8A,
F ~
Maxwell's
field
ik
where
'
dx t
dxk
the four-potential, the first set of Maxwell's equations (13)
identically satisfied. Now, consider the invariant integral
i
is
is
(97)
L-JjSfdS
12
with
L
l
022
uc
sl
A-
fi
A k(98)
integrated over a certain domain fi in (3+l)-space. s t is the four-current density and /x is the invariant mass density in the rest system.
In the
first
and
si
},
L we
8L
JJL
now
place,
we
shall
A^x)
for
which 8A l
the functions
an arbitrary variation 8A i of
at the boundary of the domain
this variation. For the variation of
consider
being kept constant by
then get by partial integration, remembering that 8
=
J
^^
=,
(J
(99)
The condition
for
SL
(100)
any variation of the kind considered then leads to the equations
to the second set of Maxwell's equations (16).
Next we shall consider a variation in which the A l are kept constant
i.e.
H. Weyl, Raum-Zeit-Materie,
(1909).
Berlin, 1918. See also
M. Born, Ann.
d.
Phys. 28, 571
ELECTRODYNAMICS IN VACUUM
158
59
V,
while the time tracks of the matter are varied. This means that only the
last two terms in (98) are affected by the variation. Let us fix our atten-
on a definite infinitesimal piece of matter with volume dV and
consider the infinitely thin tube of time tracks of this element of matter.
If r is the proper time, and x
^(T) the space-time coordinates of this
material particle, we have
tion
~>
<r<
'
"j,
dr
We now choose the domain
rl
_ / dx
__ pPUi
__ dx,
^l
dr
in (97) as that part of this tube for
which
and consider an arbitrary variation of the time track satisfyfor r ~ r l and T -^ TO. Since
r2
ing the condition 8a\
where dV
is
the rest volume of the particle,
r
I
uc 2
/^j
rfX
we have
dx
dV
<
-u
rfFc 3 dr
)
/T
T>
^
Here rfm
JLC
and
rfF
rf^
T.J
rf P
</T
-r,
3
(
/w
rf
(101)
Tl
TI
dV
represent the total rest mass and
which are constant
total charge of the particle, respectively, quantities
along the tube.
From the definition of dr
we get for the
variation of the second integral
by a variation SO^T) of the kind considered
T2
T2
l
dr
7 r T
/*
-=
<'
8x dr
(102)
dr
91, p. 244).
(see
Further, using
we
dr
dr
f l^
'
dr
dx k dr
tix
'
get
5 f A
o
\
^x^
f
l
i
dr
dr
d>
^
ox,.
k
xi
A
^ A,
dr
J \dx k
-4-
d8x
dr
j
dr
TI
==
f ldA -kTT
U
I
/^'y*
\ca^
'
ik
dA\
-!
r/T
ar
JJ fir (]T
k
i
s
o^.aT
f
I
l&A k
(-7^
y
\
f)
\ox l
tiA\
-
f} fY
>
TJ
^IcAo^jaT
i
o^kl
'
(\(\ \\
\
"'/
ELECTRODYNAMICS IN VACUUM
59
V,
From
and
(98), (101), (102),
this case
8L
get for the variation of
in
Uk -^
^F
lk
we thus
(103)
159
(104)
Thus the condition 8L
for
any variation of the kind considered leads
to the equation
i.e.
just the equations of
motion
Tr
(96) of
charged incoherent matter.
The electromagnetic energy -momentum tensor
We shall now show that by virtue of Maxwell's equations
60.
the four-
force density (93) can be written in the form of the divergence of a
symmetrical tensor. Introducing (16) into the expression (93) we get
Further,
we have
l8F
where we have made use of ( 1 3) and of the antisymmetry of Flk Thus we
~
may write ji in the form
.
/.
-,
,,0.)
('06)
According to the rules of tensor calculus, SlA
S,*
and
it satisfies
is
a tensor,
it is
symmetric,
(107)
-S*t,
the identity
0.
we
(108)
calculation the following
in
tensor
terms of the electric and
of
this
components
vectors:
field
__ __/
magnetic
Using
(11) in (106)
get
by a simple
expressions for the
where
is
lK
tf l tf(f +fl
,#,-i(*+#)S w
Maxwell's stress tensor; further,
(HO)
ELECTRODYNAMICS IN VACUUM
160
where the vector S with components S
is
Poynting's vector,
= c(ExH).
W = $(E*+H
where
60
i.e.
(112)
S44 =-JF,
Finally,
is
V,
(113)
2
(114)
the electromagnetic field energy density.
4 may then be written
(105) with i
The equation
/4
= -(f.u)
*"
ij.
--divS
+
dW
=a!
(f.u)+divS +
^=
(115)
0,
ct
and S are interpreted
which expresses the energy conservation law if
as field energy density and current density, respectively. If we integrate
(115) over a finite volume V in space enclosed by a fixed surface or we get,
by means of Gauss's theorem,
-| J W dV
V
J
a
Sn da+
(f
u) dV,
(116)
where Sn is the component of S along the outward normal of the surface
element da. The decrease in field energy inside V per unit time is thus
equal to the outward flux of field energy through the surface a plus the
total work done on the matter inside V by the electromagnetic forces.
For i = 1,2,3 (105) gives
f
MI*
(S\
t'-wrwr
field the last term is zero and (1 17) is then exactly
Maxwell 's expression for the force density in a substance with = ^ =1.
The present deduction is due to Minkowski.f As pointed out by Abra-
In the case of a static
ham, J the vector
(118)
must be interpreted as electromagnetic momentum density if we want to
have conservation of momentum for a closed system. For if we integrate
(117) over the interior of a closed surface <r which contains the whole
system we get
f
t
H. Minkowski, see ref., Chap. IV, p. 136.
M. Abraham, Ann. d. Phys. 10, 105 (1903); Abraham-Becker, Theone der Elektn-
zitat, vol. II,
6th ed., Leipzig, 1933.
ELECTRODYNAMICS IN VACUUM
60
V,
p\
__i*
since
J fa K
dV by partial integrations can be transformed into an integral
over the surface a where the
hand
field
and therefore
also
LK is
side of (119) represents the total force exerted
equal to the increase
is
161
tum. Thus (119)
dGm /dt per unit
zero.
The
left-
on the matter and
time of the mechanical
momen-
be written
may
In order to obtain a constant total momentum we have thus to assume an
electromagnetic momentum J g dv besides the mechanical momentum.
It is true that from this argument we can only conclude that the electro2
magnetic momentum density must be S/c -f-(7, where C is a constant,
but since g must vanish simultaneously with the field the constant C
must be zero.
Now
of the electromagnetic
energy by the equation
we can
defining the velocity of propagation
w="
w'
write for the electromagnetic
momentum
density
analogy with (IV. 233), holding for the mechanical momentum
density. This analogy is close only in the case where the velocity w
For a plane polarized
defined by (120) satisfies the condition w
c.
in
electromagnetic wave we have (E.H)
.
IS|
<
and
cEH
E=
//;
thus
we get
the energy in such a wave travels with the velocity of light.
field of a charged particle in arbitrary motion we have, on
account of (61) and (62),
i.e.
For the
(E.H)
i
w=
61.
The
total
and
__
cEH
E/H
<x^l,
-- ^^
2ca
<^
c.
energy -momentum tensor
If we use the expressions (105) for the electromagnetic four-force
density in the equations of motion (IV. 229), the laws of conservation
3595.80
ELECTRODYNAMICS IN VACUUM
162
of energy and
magnetic
field
momentum
for the total
V,
61
system of matter and electro-
take the form
0,
O lk
+K
Tlk =
where
(122)
(123)
lk
the total energy-momentum tensor of the system. The components
of this tensor are obtained from the components of 6 lk and$,^ given in
is
51
and
60.
We
have
IK
.^)
where
iK is
/i 9j\
(124)
*
'
Maxwell's stress tensor.
= -^-T,. u 4 c(E X H)
u /c~)
where
,/(!
is
the to\ 'il energy current density. Further,
(126)
is
the total
Finally
momentum
T44
we have
where
is
density of the system.
the total energy density.
The sum of the diagonal components of this tensor
(108)
and
(IV. 198),
(127)
7/,
is,
on account of
VI
GENERAL CLOSED SYSTEMS. MECHANICS OF
ELASTIC CONTINUA. FIELD THEORY
62. Definition of a closed system
IN Chapter V we have treated the case of incoherent charged matter
under the influence of electromagnetic forces. We saw that the fourvector fi describing these forces could be written as the four-dimensional
divergence of a tensor which itself is a function of the field variables
describing the electromagnetic field. As the principle of relativity requires that all signals are propagated with a velocity smaller than or
equal to c, it is impossible to maintain the Newtonian idea of forces
acting instantaneously over finite distances in space It seems necessary
to assume that all forces between material bodies are transmitted by
means of an intermediary field in a way similar to the case of electromagnetic forces. It is therefore generally assumed that all types of
forces can be described by a four-force density ft which is the divergence
of a certain tensor Slk depending on the field variables of the intermediary
fields by analogy with (V. 105). For the total system of matter and
fields we then get, in the same way as in
61, the Jaws of conservation
of energy and momentum in the form
'Ir^
where
The
Tlk
is
physical
(1)
'
energy-momentum tensor of the closed system.
meaning of the components Tl4 and T4l is the same as in
the total
(V 125, 126, 127),
i.e.
= -S
4ti"
T*
where S
is
(2)^
*
the energy current density,
T =
14
icg L
T44
-A,
(3)
where g and h represent the total momentum density and energy density,
respectively. This should hold for any closed physical system, and hence
also for elastic bodies, if the elastic stresses and energies are included.
The equations
(1) for
1, 2,
may now
which represents the law of conservation of
be written
momentum
in differential
GENERAL CLOSED SYSTEMS
164
TIK
form.
is
Similarly, the equation (1) with
tion for energy.
total
The
current tensor
0.
(5)
ct
energy-momentum tensor of a
au
62
4 represents the continuity equa-
^=
divS +
The
momentum
called the stress tensor or the
51, p. 137).
(cf
VI,
spatial pait of this equation,
closed system
must be sym-
*i,.
TLK
i.e
-~
TKL
is
essential for the
51 and
validity of the conservation law of angular momentum (see
f)3), and if this law is to hold in every system of inertia, the equation (6)
must
also be valid for the space-time
components,
TA = Ttl
or,
on account of
(2)
and
e.
we must also have
(7)
(3),
S/c*
(8)
Defining the velocity of propagation u* of the energy
u*
-^
by
S/h
(9)
as in the case of the electromagnetic energy (V 120), the equation (7) or
(8)
can be written
which
is
formally analogous to the equation (IV. 233) for the mechanical
density and thus shows that the energy density h corre-
momentum
sponds to a mass density
7/2
should be remarked, however, that the velocity u* defined by (9) may
be larger than c and even if ?/* < c the transformation properties of the
velocity u* by Lorentz transformations will not in general be in accordTt
ance with the transformation equations
material particle.
corresponding to
Introducing in every
we
(l[.
55) lor the velocity of a
also be negative, thus
The energy density h may
a negative mass density /?/ 2
.
system of coordinates four quantities
>V?
where
have, on account of
'
(13)
(5),
c'.r,
in every
system of coordinates The quantities >S do not,
t
ho\\ ever, trans-
form
h
MECHANICS OF ELASTIC CONTINUA
62
VI,
like the
>
and u*
components of a
<
Let us now assume that
four- vector.
c, i.e.
8,S
S 2 -c% 2 <0.
For every system of coordinates we
will
(14)
may then define
analogous to the four-velocity of a particle,
These quantities
165
four quantities
/7*,
by
transform like the components of a tune- like fourvelocity u* transforms like a particle
when and only when the
velocity. We shall now find the
vector
Tlk
for this to be
condition to be imposed on the tensor
Since
the case.
77*
__
Now
consider an infinitesimal Lorentz transformation
x(
=~-
+ c lk x k
tk
(17)
kl
connecting the space-time coordinates of two systems of inertia
the transformation laws for a tensor \ve then get
S and
S'.
From
thus, on account of (12),
e
'
I
^'
+ ,/**>
'
(17')
Expanding
in
terms of the infinitesimal quantities
terms of order higher than the
e tk
we
get, neglecting
first,
77*
77*77 u
u
*ki
i
Thus, in order that U* shall transform like a four-vector the tensor
must, for i
1, 2, 3 and for all values of &, satisfy the condition
Rlk
^T + T^. =
Tlk
ik
Q.
(19)
GENERAL CLOSED SYSTEMS
166
For
4 the equation (19)
identically satisfied.
however, also sufficient, for, if (19) is satisfied in S,
i
is
I/?'--
lfi
and
will
then transform
like
may
lK
V*k
This condition
is,
we have
(20)
a tensor by the transformation from 8 to 8'
8'. Since a finite Lorcntz trans-
be valid in
(19) will therefore also
formation
U* +
62
VI,
be composed of an
infinite
number
of infinitesimal
Lorentz transformations, (19) represents the general condition which
Tlk must satisfy in order that the velocity of the energy u* shall transform
a particle velocity
In general, the energy -momentum tensor of a physical system will
There are, however, a few
not, of course, satisfy the condition (19)
like
cases in which this condition
must be
physical reasons. In
the trivial case of a system consisting of incoherent matter without any
9 1k
Uk i e.
external forces, for instance, we have Tlk
fulfilled for
^U
In this case the propagation velocity of the energy is, of course, identical
with the velocity oi the matter For an elastic body, however, the
condition (19) will in general not be fulfilled, as we shall see in 65.
In a later section ( 70) we shall meet another important case in which
the condition (19) must be satisfied lor physical reasons.
63.
Four-momentum and angular momentum
four-tensor for a
closed system
Putting
(j,
T, 4
T-.
[g,
ic
the symmetry condition
(7)
may
17*
h\
(21)
be written
--
!A
(22)
and, on account
<>t
(13),
we have
also
(23)'
V
t
It
should be lemembcied that the quantities g and
t
are not four-
vectors.
Let us
ponents
multiply
now assume that the system
Tlk
(1)
considered
is finite
so that
all
are zero outside a certain region in physical space.
by
(h\(Lr,djc 3
comIf
we
and integrate over the whole physical space
MECHANICS OF ELASTIC CONTINUA
63
VI,
for constant
of
# 4 the
,
first
three terms in
TIK with respect to the space
(1),
167
which are partial derivatives
coordinates X K will give zero. Hence
,
we
which shows that the four quantities
(24)
and // represent the total linear momentum and
the total energy of the system respectively. Another consequence of
(1) is that the quantities G transform like the components of a four-
are constant in time
four-momentum vector. This is seen in the following way.
Let a be an arbitrary but constant four- vector. The four- vector
vector. the
l
bk
will
a.Ttk
(25)
=-
{26)
then satisfy the equation
2-k
dx
on account of
If
(1).
we multiply
(26)
by
d^l
(Lr^djc^djc^djc^
and integrate over a
finite
region
in (3
+ l)-space, we get by means of the generalized Gauss
theorem (IV. 191)
0= f^rfS =
J
(
rA
[b k
dVk
(27)
J
11
where Q is the three-dimensional boundary of the region H. Since we
have to deal with a finite system, the region in (3+ l)-space where Tlk
is different from zero, i e. the time track of the system, will have the form
of a tube with a finite cross-section in the space-like directions
Consider two arbitrary coordinate systems 8 and $' in (3+l)-space
and two hyperplanes I2 t and 11, defined by the conditions
jc^
and
constant
constant,
For the region
domain bounded by the hyperplanes iij, } 2 and by a
respectively, the values of the constants being arbitrary.
X we now choose
cylindrical surface !3 3 enclosing the tube in
The three-dimensional hypersurface
Q, is
which
Tlk ^
(Fig. 15)
thus composed of the parts
GENERAL CLOSED SYSTEMS
168
I2 1? I2 2
and (2 3 The contribution
to the integral b k
J
12 3 is zero,
since
Tlk and
h dVk
bk
dVk
we
63
dVk from the cylinder
get
=-
(28)
ila
i2i
The two integrals in
on Q 3
b k are zero
n
Thus,
VI,
and may thus be calculated in any
(28) are invariant
We choose to calculate the first integral in S and
system of coordinates
= const
- const
FIG
15.
the second in 8' If we suppose the events on D 2 to be later in time
than the events on 1^, the outward normal on I2j is pointing in the
direction of the negative time axis and, according to (IV. 191'), the
components of dVk are
dV,H
Thus we
~--
4-i dx*1 dx^dx^}
J
(0,
0, 0,
\
7.
T,
get
HI
where
constant in this integration
,r
4 is kept
In the second integral we have similarly
hence
f
J
Using
h,A.
dK.K
(29)
and
'
b,h.
rt
dV
/w
f
i
(30) in (28)
64
we
dV
O 4V
4 (.r 1'
1?
a*o,
-> a
get
=~
b'4
dV',
(30)
or,
MECHANICS OF ELASTIC CONTINUA
63
VI,
by means of
and
(25), (21),
al
G =
l
169
(24),
a\ G(
= invariant.
(31)
Since this equation must hold for an arbitrary constant vector a t it
Gl and G\ is given by the transforma-
follows that the connexion between
tion equation for a vector:
(31')
,*<?*
For the validity of this proof of the vector character of Gl it is essential
Tlk is everywhere regular. If Tlk has a singularity along a certain
world line, we would have to exclude this line from the region S by a
certain surface 11 4 which then would give a contribution to the integral
that
fb k dVk m(21).
a vector, Gl GT is an invariant, and we may
total proper mass J/ of the system by the equation
Since
is
fft
=-Jfc
by analogy with (IV 51), holding
By means of (1) and (6) we get
now
define the
(32)
for a material particle.
further
A (XtTu-x^) - S ^-8^, - T -T -
0.
lk
kl
f,
(33)
dXj
Integrating this equation over the whole physical space
arguments similar to those used before
which shows that the
find
by
six quantities
M - J (x,g
lk
we
-x k9l dV
~=
-M
k%
are constant in time. This result depends essentially on the
(34)
symmetry of
the energy-momentum tensor. By a method similar to that used in the
proof of the vector character of G it now follows from (1) that the
l
like the components of an antisymmetrical
quantities
lk transform
tensor: the angular momentum four-tensor with respect to the arbitrary
origin of the coordinate system.
The
spatial part of this tensor
IK is,
according to (IV. 98), dual to an
axial vector
(35)
which
is
equal to the constant total angular
closed system.
momentum
vector of the
GENERAL CLOSED SYSTEMS
170
VI,
64
Centre of massf
64.
We may assume
that
for
any physical system
is
a time-like vector
defined by (32)
30) In this case it is
(cf
always possible to find a system of inertia $, the 'rest system', in which
-the total linear momentum
0, so that on account of (32) we have
so that
a real quantity
is
{}
for
he components of
8\
{
in
G!
(0,0,0,iJ/
(36)
c)
Exactly as in the case of a material particle the velocity u of the rest
system ^S relative to *S is then
c*G/H
--
(37)
Tn Newtonian mechanics the centre of mass of a physical system with
the mass density ^ -- /u,(x,/) is defined as a point with the coordinate
_l
vector
M (x,0 xrfr,
(38)
In lelativistic
fidV is the total mass of the system
is
connected
the
mechanics the mass density
with
energy density by the
equation (II) We ma\ then define the eentie of mass by the equations
As we shall see in a moment, the point defined in this way
(3<S) and (11)
will, however, in geneial depend on the system of cooidmates which is
\vheie
used
m the evaluation of the integrals in (38),
each system of inertia
which depends on the system >V In /S
own
centre of mass (\tf)
-its coordinate vector X(G (A>))
X(/S )
(24), defined by
A has
r
its
is,
according to
(38), (11),
and
xrfl
From
(23)
(39)
m every system of coordinates
we now get
'
tr
)(r/ '
* )
>
-a:A^
- Ji
(
a 8 ik -
u
,
and by integiation ovei the whole 3-space
constant in time this equation for k -~ 1, 2, 3 shows that the
cent e of mass defined by (39) is moving with the constant velocity
2
G/// re kit iv e to $, i e. with the same velocity u as the rest system S.
Thus all the different mass centres C(ti) aie at lest in the system $
Since
// is
<
f"
\(fi>l
Soi
Kokkcr, Rcldtn
1)
<1
A,
M H L
\thttnti,
No
.3
14,
(
(1
,~>4()
(1{)3 )),
('
C <U*
Hoy Sot A, 195, 02
)I9), ()
Ii>co, 7'/w
(jionnigiMi, 1920, p 170, A Papapetrou, I'raktikci
Mollor, Cotnm Dublin lust for Advan< cd Studies,
Heaunyaul, Kdatintt' tefit/etnte, rhap i\ Pails, 1949,
itnt^thcoi u
(1948)
VI,
MECHANICS OF ELASTIC CONTINUA
64
171
One of the
C(S
centres of mass plays a distinguished role, viz. the point
which is the centre of mass in the rest system itself, it may
Q
)
be called the proper centre of mass
If
(X, A" 4 ) are the space-time
coordinates of the proper centre of mass C in an arbitrary system of
coordinates, the
l
X^T) will be linear functions of the proper time
L
(}
r of this point.
if
Further,
U -
dXl
dr
denotes the four-velocity of
(7?
we have, on account
G,~M
The dependence of G on the
l
of (37),
(40)
velocity of the proper centre of mass
the same as for a material particle
Now denoting the relative angular
momentum
respect to the proper centre of mass C
four-tensor
is
m lk
thus
with
by
'.-XJyL-^-X^] dV - M -(XJ!k -X k G,),
tk
we
get,
by
differentiation with respect to the vanables
on account of
The two space vectors
(40).
m=
-in
Wai,
(?/?,,
f >,'
>C)
we
(41),
and
r,
n defined by
therefore in (41) choose
get
((x-X)Xg)r/l'
ci-A'i
(43)
We may
and, by means of (42) and
W 10
(42)
-=
are thus constants of the motion.
x4
m and
.*
(41)
a
J e
"
'
^4
Xt
y*
the relative angular momentum vector with respect to the
centre
of mass, the inner angular momentum, while
n/c is the
proper
moment of mass with respect to the same point Solving the second
Thus rn
is
equation (43) with respect to
~>/J
hx dV
X we
get
on
(44)
where X($) and X($)
X are simultaneous coordinate vectors of the
centre of mass C(ft) in X and the proper centre of mass C($)
From
(44)
we
see that the different centres will coincide only if n,
GENERAL CLOSED SYSTEMS
172
and therefore
m lk
zero in every system of inertia, i e
considered has no inner angular momentum If (44)
rest
system
get
n
since
by
definition
--
is
Thus
mass
seen from (36)
,>
{j
'-
when equation
get,
written in the
(45)
is
equivalent to the
n
u
(46) is wiitten in the
way
system
iS'
that the proper centre of
system
choose the same orientation of the spatial axes in 8 as in tf
of the transformation equations (IV. 81') for an anti-
by means
symmetiical tensor, on account of
(45),
the velocity of 8 relative to
momentum
vector in the rest system
angular
where v
is
0,
X(tf). This condition
-----
this equation expresses in a co variant
is the centre of mass in its own rest
When we
we
w?4 =
or
covariant equation
as
when the system
is
we
xS
64
VI,
is
AS
and
is
the inner
The difference between simultaneous positions of the centre of mass
in 8 and the proper centre of mass is, according to (44) and (47), gi\en
by the time-independent space vector
a(,<7)
- XOS')-X
-en///
.=
(mxv)/3/
c2
(48)
where we have used the relations
following fiom (40).
Since the transformation from
$ to Nis given by a
Lorentz transforma-
and since a is perpendicular to the relative velocity
the distance between the two centres, mentioned above, in the rest
tion without rotation,
v,
system *S is also given by (48).
In the lest system $ all mass centres f(#) obtained by varying $ or v
in (48) foim a two-dimensional circular disk perpendicular to the
angular momentum vector
and with radius
with centre at the proper centre of mass
,
"
In the non-relativistic limit
we
c -> oo
ft!
<
JU Q C
49 '
the radius of the disk tends to zero and
are left with one mass centre only, the Newtonian centre of gravity,
in general not one centre of mass but
but in the relativistic case we have
the disk of mass centres mentioned above, the centre of which
is
the
MECHANICS OF ELASTIC CONTINUA
64
VI,
173
proper centre of mass. Only if the system has no inner angular momentum is the radius (49) of the disk zero. It is true that the radius (49) for
all macroscopic systems is very small compared with the dimensions of
the systems For the earth, for instance, we have
/Varth
10 metrcs
(50)
'
JM0C
For systems of atomic dimensions, however, the radius of the disk of
mass cent es may be comparable \\ith the dimensions of the system.
i
Fiom
we can draw
the above considerations
a certain conclusion
regarding the dimensions oi a system with given inner angular momenand proper mass
Consider an arbitrary physical system
tum
which in the rest system 8 lies entirely inside a sphere with centre at
{}
the proper centre of mass C and radius r, i.e. a system for which aU
components of the energy -momentum tensor are zero outside this
sphere.
If
where in
all
centres
we
further assume that the energy density h is positive everysystems of inertia, it is clear that the whole of the disk of mass
must
lie
inside the sphere; for if
we
consider an arbitrary point,
say C(R), on the disk, this point will in the system of coordinates H be a
centre of mass, and since h is positive it must then lie inside the physical
system
We
thus get
r >>
'
(51)
'J5 c
3/
Thus, a system with positive energy density and u ith a given inner angular
momentum m and a given rest mass
must always have a Jinite extension
()
(}
in accordance with (5
positive in all
65.
L)
systems
the system
It
The fundamental equations
Jn
Chapter IV,
is
smaller, h cannot be everywhere
of inertia
50 51
matter under the influence
we have
of
mechanics
in elastic continua
treated the mechanics of incoherent
of given external forces.
We
shall
now
con-
body with no external foices The sole forces
acting in the body urc then the clastic torces between neighbouring
parts of the matter due to the deformation of the matter. We have thus
sider the case of
an
clastic
to deal with a closed system \vhieh
is a special case of the general
the
considered
in
and
62,
systems
equations (l)-(l 1 must be valid for
the total energy-momentum tensor Tlk of this mechanical system. The
)
mechanical energy-momentum tensor has, however, especially simple
properties which we shall now establish
(Consider an infinitesimal lace element da with a directed normal
defined by a unit vector n, at a definite point
in space
The matter on
GENERAL CLOSED SYSTEMS
VI,
65
either side of this face element experiences a force which is proportional
to da. The force acting on the side to which the normal points will be
and reaction are equal, the force on the
(1)
n (2) n (3) are unit
t(n) da If n
n) da must then be
t(
the directions of the Cartesian axes, we have
called t(n) da, and, since action
other side
vectors in
t(n)
(52)
n% are the components of the unit vector n The equation
obtained by a consideration of the infinitesimal piece of matter
where w 1?
(52) is
??
2,
Fio
16.
contained in the pyramid abcp of Fig 6 If da is the area of the triangle
abc, the area of the triangles pbc, yea, and pab are n^da, n^da, n^da,
respectively, and the total elastic force on this piece of matter will be
1
-t(n)
rf<j-ht(n< >)w 3
da
This force must be equal to the change of momentum of the matter per
unit time, i.e. d(&oV)/dt, where g is the momentum density and oV
the volume of the pyramid In the limit of an infinitely small volume,
oV tends to zero faster than da, therefore
MECHANICS OF ELASTIC CONTINUA
65
VI,
which leads immediately to the equation
vectors t(n (K) ) are denoted
by
iK ,
t,(n)
lK
(52)
may
lK
If the
components of the
be written
nK
(53)
n K are the components of space vectors, the quantities
must transform like the components of a space tensor by rotations of
Since
t
(52)
175
(n) an(i
the Cartesian axes.
LK
is
the elastic stress tensor, sometimes called the
TLK of the total energy -
relative stress tensor, in contrast to the space part
momentum
The
a
is
tensor
Tlk
which
total elastic force
now
equal to
is
called the absolute stress tensor |
acting on the matter inside a closed surface
Jt(n)rf<r,
a
where n
is
the outward normal to the surface element
The components
of this force may,
theorem (IV. 192), be written
L
l(t
rf ff
by means
= -
r/cr.
of (53) and Gauss's
(54)
j^'dV,
Si
where the integration on the right-hand side is extended over the interior
of the closed surface a Thus \ve can define an elastic force density f
such that
*',- //,</!',
(55)
1*2
and a comparison of r>4) and (55) shows that the clastic force density
and the relative stress tensor are connected by the equation
(
/.= -The motion
is
*)
of an infinitesimal piece of matter with the
the equations of motion
volume 8V
now determined by
(57)
^Sn^Sr.---^!',
where g
is
derivative
momentum density and d/dl denotes the substantial time
By means of (IV 201) and (IV. 204) we get
the
68 >
Ann d
\on Lane, Die
Relativitatsthcone (3rd ed
Fht/a 35, 524 (1911).
Braunschweig 1919), vol
i,
29;
GENERAL CLOSED SYSTEMS
176
65
VI,
where the U K are the components of the velocity u of the matter at the
place considered. From (57) and (58) we then get
On
0.
(g^tc^uc)
JLi-j
(59)
the other hand, the law of conservation of
we obtain the
thus
momentum
is
also
following connexion between the
expressed by (4);
absolute and the relative stress tensors
TLK =
lK
+g
uK
(60)
In order to find an explicit expression for the momentum density
connexion (8) between g and the energy flux S:
we
shall use the
The
total
work done by
surface a per unit time
S/c
(61)
on the matter inside a closed
elastic forces
is
f (t(n).u)dcj
--.
11
where the integration in the last integral is extended over the interior 12
of the surface a. The work done on an infinitesimal piece of matter of
volume
is thus
f](
A = _ iT-'^sr.
^
(62)
This must bo equal to the increase per unit time of the energy inside
which
-~
rfr
/
SF
is
K
*'
(63)'
v
being the total energy density including the elastic energy. Thus
get from
(63)
and
(02)
?
h
i
01
A comparison of (5) and
+ ,?-(l
(.1 K
(64)
+ vJJ =
0.
(64)
shows that the total energy
flux
is
t) is
given by
r
S-Auf(u.t),
where (u
we
(<> >)
a space vector with components (u.t)^ --- u t lK
Thus,
hu there is an extra transport of energy
L
besides the convection current
due to the work done by the
get for the total
momentum
g_
elastic forces
From
(61)
and
(11)
we then
density
_. + ("'.
,66,
MECHANICS OF ELASTIC CONTINUA
65
VI,
where
/x
h/c
177
the total mass density including the mass of the elastic
of the last term in (66) the momentum density
is
On account
energy
vector has not in general the same direction as the direction of motion
of the matter hence
.
9i
U K-9* u
Since the law of conservation of angular momentum requires
we
63),
(see
get from
IIK-IKI
i
~</c
Only
_
--
U K \^K U
/O
l
iK
/po
-*-
where h
IK
is
7^ t
(-(U.t) U K +(U.t) K U
L
)/C*
0,
not symmetrical
momentary
system A^ of the matter at the point conhave u
and thus, on account of (60), (65), and (66),
in the
we
-----
(60)
e the relative stress tensor
sidered,
TIK
is
rest
._ 7H)
__ /O
1
HI
tfO
*\
KI>
__ dO
--
fel
M
_- U
rfiQ
L 14
710
L 44
>
~ _7,0
II
V'
(l\>7\
9
the rest energy density.
The mechanical energy-momentum tensor
satisfies
the equation
Ttk T?k = -A U
(68)
t ,
the four-velocity of the matter The validity of the co variant
equation (68) follows at once from (67) if it is written down in the rest
The relation (68) is characteristic of
(0, 0, 0,tr).
system, where U**
where
\ is
a pure mechanical energy-momentum tensor and contains equation
(IV. 246) as a special case. If we multiply (68) by U we get the followt
ing expression for the invariant rest energy density
For
or,
&
= ~
1, 2, 3,
by means of
l^Tlk Uk jc*.
(68) gives
and (TV
(60), (3),
39),
(^+^vK-c
Solving with respect to
(J L
&
where
/x
h Q /c 2
(69)
we
get a
-t
-AX.
new expression
^u + (l/e*)(t.)
r~ T
for the
"
(68) for
iK
uK
4 gives similarly, on account of (70),
/z,
3595.60
u/
the rest energy density and (t. u) is a space vector
u K By means of (IV. 39), (70) may also be written
llc
--4
with (u.t.u)
'
is
with components
The equation
momentum
XT
-j-(u.t. u)/c
(71)
GENERAL CLOSED SYSTEMS
178
(72)
may
The equation
-h - + M
----
Ut -
C/4
V^ U
fS
(73)
(69) gives similarly
The right-hand
identical with (72)
(74)
.u),
ris
65
also be written
Tu
which
VI,
side of (74)
is
thus an
invariant scalar
If
we
divide (74)
c2
by
we
get
which is a generalization of the equation (IV 199) valid
matter
Comparing the two expressions (66) and (70) for g we
for incoherent
get,
of (75), the following identity for the relative stress tensor
(l-w 2
)(ii.t)
/<'
(t.u)-u(u
t.u)/r
by means
lK
(76)
A closed system may always be divided into non-closed sub-systems in
an
infinite
number
momentum
of
ways corresponding
where
is
^U
B, k
the kinetic
(IV. 39)
we
to a division of the energywe may write
tensor into separate parts. For instance,
Vk
energy-momentum tensor
(7S)
By means
of (60), (71), and
get
Thus we get
and (73)
for the
components of the tensor
"
'
(^
On account
__
T T TT
rjJ
,,
t.A
K,^
from
^A
satisfies
the condition
a relation which also follows directly from the expression
In the rest system we get from (80)
-
(71),
(SO)
of (68) the tensor
-SSc
(70),
C,
'^4
^,
-=
(HO).
(82)
MECHANICS OF ELASTIC CONTINUA
63
VI,
olrtst
Putting
the equations
/,
(1)
may
179
aSffA
(83)
dx k
be written in the form
elast
/,
Tt should be noted that the force density
229)
defined by (S3), is not in general identical with the elastic force f
defined by (56) A simple calculation shows that (// hlst f / ) which, as
shown in 50, means that the proper mass of the system is not conserved
analogous to (IV
f'
last
This is also natural, since
/x,
includes the mass coi responding to the elastic
of the elastic forces
While
energy which changes under the influence
the tensor O lk satisfies the condition
the total mechanical energ\ momentum tensor Ttk will in general not satisfy this condition
The relative stress tensor t LK is connected with the internal deformation
of the matter
In the rest system
(19),
/S
this
connexion
is
given by the
equations of the usual theory of elasticity, thus for small deformations
it is given by Hooke's law
By means of the transformation properties
of the stress tensor this connexion can be established in
any system of
ineitia |
subjected also to external non-mechanical foices
described by a four-force density f[ xt w e have instead of (1)
If the elastic
is
body
(S5)
or
by means of
66.
(77)
and
Transformation
(83)
of elastic stress,
momentum
density,
and
energy density
Let us assume the spatial axes in the coordinate systems N and
to
iS
-have the same orientation. Since the velocity of >S relative to *S is the
same as the velocity u of the matter at the point considered, the trans-
formation coefficients a lk are given by (IV 125),
/i/
rti
e
i
(87)
Heiglot/.,
Ann
I'lnjs.
36, 493 (1911).
GENERAL CLOSED SYSTEMS
VI,
66
the transformation equations of a tensor in connexion with (67)
we
180
From
gCt
-,
tk
For
T*^*^ -~ f^w^-h'w*.
(88)
4 this gives
(89)
comparison
of (89)
and
(72)
(u.t.u)
For
i --=
or, in
--
1, 2, 3,
4,
we
(u.t.u).
(90)
get from (88), by
means of
(87),
\ector and tensor notation,
In the same
and
(<>0)
shows that we must have
way we
get from (88) with
c,
fc
--
K-,
and from
(1)1)
(93)
Intiodueing the notation a O b for the direct product of the space
vectors a and b, \vhich is a tensor with components a b K1 the formula
L
may
(93)
also be written in thiee-dimensional tensor notation
U-
yu
y
(94)
By means
T
oft/his
(t
u)
---
tiansformation equation
it is
easily verified that the
remembering that t tK is symmetrical, i e.
(u t), a simple calculation shows that the expression (94) ior t
elation (90)
is
valid, further
(}
accordance A\ith the equation (76).
In the special case where u
(u,0, 0),
is 111
i.e.
where the motion of the
matter at the point consideied is parallel to the #-axis, the transformation
VI,
MECHANICS OF ELASTIC CONT1NUA
66
equations
(89), (92),
1
-*
and
reduce to
(93)
iQ
'
yjc
** '
'UU^
i///
~"
'
181
U**
UZ
""'
67. Perfect fluids
In a perfect fluid the force t(n) on a face element
parallel to n,
i.e.
t(n)
is
the normal pressure
where p
is
the normal pressure.
In the rest
P
or
e
(93) in this case
^LK
P
the normal pressure
From
(SO)
is
(53)
system we have in particular
The transformation equations
normal n
Thus we get from
where p
\\ith
and
(97)
is
~~
P^
~~
P>
educe to
^IK>
(100)
an invariant scalar f
we now
get for a perfect iluid
(101)
(102)
the invariant pressure. This expression for 8 lk also
follows from the fact that (102) is identical with (82) in the rest
where
^>
pQ
is
system.
t
M. Planck,
Berl.
Ber
p.
542 (1907), Ann.
d.
Fhys 76,
(1908).
(;KNKK\L CLOSKD SYSTKMS
jH2
From
follows that the piessure
of the tensor >V, A i e
it
(102)
diagonal
sum
total
and
(78),
one-third of the invariant
is
The
vi, ^GT
J'S,
eneigy-momentum tensor
toi
lo:J )
a perfect fluid
is
then, by (77)
the rest density jj? and the temper atuie which j,s given by the equation of state of the fluid.
It the fluid is subjected to external forces with the foiu -force density
The pressure
rxt
a Junction
/>" is
/>
of
the ecjuations of motion ot the fluid
may
OJlk
be written in the form (86)
*k
CXfc
-.
r2
Let us assume
>xt
i
//
r2
xk
r2
dr
dr
(105)
Pjt\
to be of the type (IV. 214)
which
satisfies
the
xt
The action of these forces will then not
identity (IV. 216), P,/;'
creation
of
to
rise
an\
pioper mass It is different, however, \Mth
give
the forces (105) ior which
=V
<)jc
--
*-
^dtv u
(106)
div u is the \ olurne dilatation in the rest system,
j)div u represents therefore the inciease in elastic potential energy density per unit
time in the rest system. JSince ^ includes the mass corresponding to
elastic potential energy,
'
c-
'ojc
thus represents the rate of creation of
proper mass density in the icst system
ming we now
Multiplying (8(5) by
the
get
equation analogous to (IV 226)
dx k
c2
f~.
which just expresses that
c2
c2
and sum-
V/
rj
dx k
represents the source density of proper mass. Only if the fluid may be
considered as incompressible do we have dUk /dx k
div u
0, and the
proper mass will be conserved.
VI,
MECHANICS OF ELASTIC CONTINUA
67
By means
of (107)
we now
183
get
(109)
/Ti
(*>T
('Ei
Thus, using (109) and (105) in
motion for a perfect fluid.
(86),
we
<
get the following equations of
^
(no)
dr
For the energy density and momentum density we get from
(89), and (97), or directly from (104),
(66), (70),
(in)
^, and u are constant throughout the elastic body, we get by
^ 2 /c 2 of the body
integration of (111) over the whole volume V = F\/(l
If /x,
G - gF =
1
(112)
From
these equations
we
see that the total
momentum and
energy,
the quantities (G, (i/c)//), do not form a four-vector in this case.
This is not in contradiction with our general result in 63, because the
e.
system is not a closed system. In order that the quantities ^t />, u can be
constant throughout the body, the fluid must be contained in a vessel,
the walls of which will act on the system with forces which are not
,
included in the energy-momentum tensor (104).
From
(112)
we
find,
(See Chapter VII.)
however,
G -**?.,.
C*
II+pV
(113)
which shows that the system has the same momentum as a particle with
The quantities
H+pV and rest energy E =
energy E
H+pV.
(114)
thus transform like the components of a four-vector.
GENERAL CLOSED SYSTEMS
184
Scalar
68.
meson
fields.
General
VI,
68
field
theory
While the iorces between the atomic nuclei and the outer electrons
are properly described by electromagnetic fields, the characteristic
short-range property of the forces between the constituent particles of
the nuclei indicates that the nuclear iorces are of an essentially non-
electromagnetic nature. In order to account for the nuclear forces
Yukawaf introduced the so-called meson fields. The simplest type of
meson
the scalar
field is
x
function
field
described by an invariant scalar field
satisfying the field equation
F(.r t )
d 2lF
W* -K T--0
2
or
a constant connected with the range ol the nuclear forces and
meson field XF is a real function of the spacetime coordinates (jr ).
Here K
is
in the case of a 'neutral'
t
Introducing the notation
the
field
F,
<F
ar
,
t
equations can also be written
^YL- K V\'
These equations
may
be derived fiorn a vanational pimciple
8
where
In
fact, if
(116)
J i(T,
TJ dZ
== 0,
^ -i^F^-f^Y
il
the variation S XF -=
of
is
(117)
(118)
).
assumed to vanish at the
ST^rJ
boundary of the arbitrary four-dimensional region of integration we
X
have, since S F,
f
J
r/X
- r
f SiWi;
I
8 F+ ^ 3 I^ dX
(^
\0T
f^
X
f
J
(119)
If
Now,
as this e\})iession
considered,
we
is
to be zero for
X
any vaiiation of F of the type
get
which are the Euler e(]uations corresponding to the vanational principle
(117). With the expression (118) for AJ (1^0) is identical with the field
equations (116).
t
Yukawa, P?oc
Math.-P/ty*. Soc Japan, 17,48(1935)
VI,
MECHANICS OF ELASTIC CONTINUA
68
The energy-momentum tensor of the
meson
scalar
field is
185
given by
Ttk = T/F.+fi 8,, = f,H;-!OT+KF') 8,,.
On account of the field equation
(116) this tensor
~m
~
the equation
is
(121)
easily seen to satisfy
-*
(122)
dx k
holding for a closed system.
The scalar field is a special
number of field vanables
case of a general field described
Ql^Qlfa) -
(#Vi), Q*(*
),
by a
(123)
-).
Suppose that the field equations aie derivable Ironi a variational
principle
where
is
SJfirfS-O,
(124)
(125)
---
A!(<^, Q\)
a certain algebraic im ariant function of the
first
derivatives
(^
d(n
~
field variables
and
their
(126)
d*i
This means that the
field
equations are of the form of the Euler equations
o
n->
1 -7 )
is su])posed to
following fiom the vaiiational principle (124). Since
be an invariant, the equations (127) will have the same form in every
system of inertia
In virtue of the
field
equations (127) the quantity
+8
is
now
'*
easily seen to satisfy the divergence relation
(129)
gi-*
In
fact,
we have
'
8xk
(128)
bQl
on account of the relation
QQ^
ar,
dQ
fjQ^,
-
ftr
following
b from (126).
CiKNKKAL CLOSK1) SYSTKMS
186
The quantity
tensor of the
When
iJ is
canonical
to
(128)
therefore be taken as the
may
VI,
68
energy-momentum
field.
an invariant,
6 lk
energy-momentum
easily seen to be a tensor. It is called the
tensor, its time component T44 being equal
is
where
the Hamiltonian density
In the case of the scalar field, the expression (128) for 6 lk reduces to
the tensor Tlk given by (121). In general, however, 0, A will not be symis
metrical and O lk will differ horn the real
divergence-free tensor
where
iA
t,,-t u
energy-momentum tensor by a
so that
-,
-(0,,-0J,
=,-
(132)
'/
*k
Belmfantej and RosenfeldJ have given a general formula for the
calculation of
/, A
As another example we consider again the case of an electromagnetic
field in
vacuum
treated in Chapter
In this case the
Qt are the components of the four-potential
field variables
and the function
ii is
\Flm b]m -
fi =-=
AA lm
where
l
/
;--
"''-tn
The Kuler
(V
10) in
ecpiations (127) then take the form of the
Maxwell equations
vacuum
-(.4,-^
or
--^-0
(134)
Together with the Lorentz condition (V. 22) which must be regarded as
an accessory condition, this gives the wave equation
Bobnfanto, Fhi^ica, 6, SH7 (1U39), ibid 7, 30,"> (1940).
KuM'nfold, Mcinvirctt dc rAiail. Roy Bclyique, 6, 30 (1940)
,T.
VI,
MECHANICS OF ELASTIC CONTINUA
68
The canonical energy-momentum tensor
0,*
This tensor
is
- I Fu
^-mM
not symmetrical and
electromagnetic energy-momentum
',.
which
satisfies
(128)
it
is
187
now
(135)
8,1
deviates from the symmetrical
106) by the term
tensor (V
- -A
(136)
the equation
a/
ftr A.
on account of (134) and of the antisymmetry of the tensor
kl .
VII
NON-CLOSED SYSTEMS. ELECTRODYNAMICS IN
DIELECTRIC AND PARAMAGNETIC SUBSTANCES. THERMODYNAMICS
General properties of non-closed systems
CLOSED system H may be divided in many ways into two non-closed
(l)
(2)
systems X and S corresponding to a, decomposition of the total energymomentum tensor Tlk into two parts
69.
In the case of electrically charged matter, T$ may, for instance, be the
mechanical energy-momentum tensor and T$ the electiomagnetic
tensor. Defining a four-vector fl by
we
get fioni
(1)
and from
(VI.
1),
wj?
Jl
BV,
the four-force density produced by the system H (2) and acting on the
(1)
Thus the fundamental equations of a non-closed system are
system
is
of the form
^ ct
-'--!?
the force acting on the system with the tensor
where f
acting on the system with the tensor N |A is then
f
t
is
Tlk The
.
force
The physical meaning of the space-time components T^ and T
the energy-momentum tensor of the non-closed system is as in (VI. 2,
of
it
i.e.
ti
=-.
-*,<?,,
T lt
icg.,
3),
TM -h\
(3)
and g now denote the energy density, energy flux,
density, respectively, of the non-closed system.
Instead of (VI. 4) and (VI 5) we now have
wheie
/>,
S,
and
momentum
C
-
/44>
(4)'
v
VII,
ELECTRODYNAMICS, THERMODYNAMICS
69
which represent the momentum and energy theorems
system, by analogy with (IV 238, 236)
In \iew of the arbitrariness in the decomposition
we must have
be symmetric, but in any case
T -Tlt
--=
tle
total linear
for a non-closed
(1), the energyhowever, not necessarily
momentum tensor of a non-closed system need,
The
189
momentum and
-OS'^-A,).
(5)
energy
of a non-closed finite system is, of course, in general not constant in time.
Integrating (2') over the whole physical space in an arbitrary system of
inertia
If
/S,
we
G (t) and
L
system in
get
G' (t
two
represent the
momentum and energy of a non-closed
between Q
different systems of inertia, the connexion
not be given by (VI. 31') This is already obvious from the
fact that there is no unique connexion between the variables t and t*
and G[
will
occurring as arguments in G and G(. But, even for a stationary system
where G and G[ are time-independent, the quantities G will not transform like the components of a four-vector (see 70). This follows at once
from the proof given in 63 for the vector character of G in the case
of a closed system For a non-closed system we would get, instead of
l
(VI. 28),
n,
and the right-hand
closed systems.
Also the angular
side of (7) will be zero only for very special non-
momentum
defined by
W
will
now be time -dependent. From
-T
x
3(
~~~ i
kl~ x k Til) __
~~
p~~
--
(2')
we
(8)
get, instead of (VI. 33),
f
f
Jk~~~ x k Ji
im
i
J-ki
rp
*ik'
OX^
Hence, by integration over the whole physical space,
J/lfc
=
J
(x t
fk -xk /.+Tta -5Tlfc ) dV.
(9)
NON-CLOSED SYSTEMS
190
Thus,
in this case,
defined
the density of the
moment
VII,
of the forces
69
has to be
by
d lk
A-ai/.-f
*,
Tkl ~-Tlk
_=.
*,
A-J-,
A+^-SV
(10)
For a non-closed system the centre of mass loses its physical importance Defining the coordinate vector of the centre of mass in the system
of inertia ti by the equation (VI 39)
X(A')-
we
get,
by means of
xrfr- 1
~J/>(x,o
(9)
with
--
i,
I-
f x</ 4 rfr,
and of
((>)
(ii)
with
4,
after
a simple calculation,
(12)
The
velocity of the centre of mass is thus not equal to c G/H as for a
closed system even if the energy-momentum tensor is s\ mmetncal This
2
severely limits the value of the centre of mass as a representative point
of the physical system.
4
In a closed
own
rest
system the
pjo/tet centic of HI
aw \\ us the centre of mass in its
We may now also for a non -closed system try to define
system
a representative point inside the system which at any time
mass
in its
momentary
rest system, the rest
different at different times
is
the centre of
systems being, of course,
closer investigation shows, however,"]*
that the representative point
is not uniquely defined by this condition.
even in a closed system there is an infinite number of points
which at any time are centres of mass in their momentary rest systems.
In
fact,
For
if
we imagine
64 to rotate with constant angular
the disk defined in
velocity
in the rest system S of the proper centre of mass, any point on the
rotating disk will be the centre of mass in its momentary rest system ( Con-
a point p which at the time considered has the radius
vector a reckoned from the centre of the disk Its \ elocity is then
sider, for instance,
hence
m^ v v
-ffiw
et C. Mollei,
Ann
--
yir
(o>
^ _
a)
=---
(m X a),
(mx(mxa)) =
|m
r2
(14)
a.
Inst Henri I'oiniate, 11,
fast-
v,
251 (1950).
(15)
VII,
ELECTRODYNAMICS, THERMODYNAMICS
69
191
comparison of (15) and (VI. 48) shows that the point p is centre of
mass in a system j$ moving with the same velocity v relative to 8 as
the point itself, i e. any point on the rotating disk is centre of mass in its
own
rest system.
In the case of a closed system it was possible, however, to single out
one point, the proper centre of mass, by the condition (VI. 40), which
means that the total linear momentum of the physical system is zero in
the lest system of the point This is not possible in the case of a nonclosed system
For if we apply the equation (12) in the momentary
rest system of one of the representative points defined above, the left-
hand
tum
side
is
zero
and the equation
will, in general,
2)
then shows that the linear momen-
not be zero in this system of inertia, and even if
moment considered, it w ill not be so at a
this should be the case at the
later time.
Thus a unique generalization of the Newtonian centre of
gravity for non-closed relativistic systems is possible only for very special
external forces (see 70) There is one important exception, however,
we shall see in Chapter X, 114 Tf the ext ernal forces are gravitational
forces, and if the system is sufficiently small, it is always possible to define
as
uniquely a proper centre of mass with
all
the properties of tho Newtonian
centre of gravity
70. Static
non -closed systems
Let again Tlk be the energy-momentum tensor of the system considered but let/, now be the four-force density produced by the system,
then, according to
(2'),
we have
in every
The system is called static if a system
of coordinates
physical variables are time-independent
G =
The system
as a whole
is
J g
r/r
system of inertia
=
[
and
if,
ti
exists in
which
all
further,
S dV -
therefore at rest in the system
(17)
$, and
since all
physical variables are time-independent in $, it is clear that also the
centre of mass in 8, as defined by (11 ), is at rest in 8
The system considered thus represents a case in which an unambiguous generalization
of the Newtonian centre of gravity is possible for a non-closed system.
As an example we may think of the electromagnetic field of charged
matter at rest in a definite coordinate system $. The tensor Tlk is then
the electromagnetic energy-momentum tensor 8 lk which for a substance
NON-CLOSED SYSTEMS
192
70
VII,
given by (V. 106), f being then the electromagnetic
four-force density acting on the charged matter.
Another simple example of a static non-closed system is a fluid con-
with
=^
/LI
is
tained in a vessel under the influence of the external pressure from the
walls of the vessel.
find the total energy and momentum in a system of inertia 8 with
Q
respect to which S is moving with the constant velocity u we may use
To
the transformation properties of a tensor and the expressions (VT. 87)
for the transformation coefficients a lk
.
Integrating the equation
^---Tl iah a mk
over the whole space,
we then
dV
and
(17),
G-
gr/r
(18)
get, using the
Lorentz formula
--
r<w.u
(19)
where T is the spatial tensor with the components T K Although G
and H are constant in time they do not transform like the components of
a four- vector In general, this may be taken as a proof that the system
considered is non-closed. For an elastic body, the equations (19) are
obtained from (VI. 92, 89) by integration, if we assume that the velocity
u is constant throughout the body. Such a system cannot therefore be
closed unless the stress tensor
If
is
of the form
* IK
(19) reduces to the
-__
m
\^W
,
<>
same equations as
71. Electrostatic systems.
everywhere in the body.
LK>
for a perfect fluid,
Classical
models
e.
(VI. 112).
of the electron
now consider in a little more detail the case of charged matter
at rest in a system of coordinates S. If
1, the tensor Slk is
^
Let us
given by (V. 106 114). Since the field
0, and E is constant in time, i e.
is
electrostatic in
8 we have
H =
0,
SK
-EE
+ l\E\* S
iK .
(21)
VII,
ELECTRODYNAMICS, THERMODYNAMICS
71
193
Let us in particular consider the case of a spherically symmetrical distribution of the electric charges. In this case the field will also be spherically
symmetrical, E being directed along the radius vector connecting the
centre of the charge distribution with the point considered. Hence
and
$'
dV
^- 61
|ET dV
8 llt
dV
S, K
Jffo 8 t(t
(22)
Using (22) m (ID) we get for the total electromagnetic momentum
and energy of a spherically symmetric charge distribution
r"
G(
"
el
Ol
7* V(i~*/<- 2l'
vl
Such a system represents a classical model of the electron, the fundamental equations of Lorentz/s electron theory being identical with
Maxwell's equations for substances with e = /t
1.
Lorentz put forwaid the idea that the mass, energy, and momentum of the electron could
be of purely electromagnetic origin, but from (23) we see that this is
impossible, f since the dependence of the electromagnetic energy on the
velocity differs from the relativistic formula (III 31) for the energy of a
particle. Since the quantities (G el (i/c)H^) do not transform like the
components of a four-vector, we have to deal with a typically non-closed
,
system, and in order to get a consistent classical picture of the electron
we must assume the existence of non-electromagnetic energies and
momenta inside the electron at least as long as Maxwell's equations are
supposed to hold throughout the whole space.
Let us now assume that the charge e is uniformly distributed over the
surface of an elastic sphere of radius a in the rest system. If n is a unit
vector in the direction of the radius vector, the solution of Maxwell's
is
equations
E =
for r
E =
> a,
<
for r
a,
=- 0,
(24)
where r is the distance from the centre of the sphere. Thus we get
from (21)
n n K Jri
1
j
el~ij
\E\ 2
for r
*<
2-4
>
dVO^
a
t
3695.80
Abraham, Phys. ZS
5, 576 (1904)
NON-CLOSED SYSTEMS
194
VII,
71
Here the charge is measured in Heaviside units and m^ is the electromagnetic contribution to the rest mass of the particle.
According to (V 1 09) the electric force per unit surface on the sphere is
e,
which must be
with the elastic force
in equilibrium
stress tensor inside the sphere
pQ
where
= ---
mechanical energy and
total
,
24
a4
2,
2(47r)
The
Thus the
elastic
must be of the form
~--^\3
momentum
can
(28)
now be
obtained from
(VI 112)
ff'""
"
7/SA
(29)
mo
""
"
""""(l-^c 2
(T-^Va)
Adding the expressions (23) and (29)
we
)"
get for the total energy and
momentum
/"^
i/
/^
(^'!
im--t-
F/
//
''
A system of that kind was used
model of the electron | Poincaro did
not specify the nature of the forces which in his model counterbalance
the electric forces in the electron he simply assumed the existence
of such forces of non-electromagnetic nature and a corresponding
energy-momentum tensor which together with the electiomagnetic
as
we should have
for a closed system.
by Pomcare
for the first time
as a
energy-momentum tensor Tlk satisfying the concharacteristic of a closed system
In contradistinction to this duahstic point of view, which requires the
tensor defines a total
dition
dTlk /d.r k
0,
quantities of a non-electromagnetic nature, MieJ
and Born advocated a unitary point of view in which only electromagnetic field variables are introduced These field variables must then
introduction of
field
satisfy equations
t
t
fc
which deviate from the Maxwell equations inside the
H. Pomcare, Rend Pal. 21, 129 (1906)
Mie,
liom, /Voc
Ann
d. Phijs
37,
Ml
(1912), 39,
Roy Hot A, 143, 440 (1934)
(1912), 40,
(1913)
KLKCTRODYNAM1CS, THERMODYNAMICS
71
VII,
electron where the field
195
strong These field equations are non-linear
and the corresponding energy-momentum tensor S lk satisfies the necesdSlk /dx k is zero.
sary condition dti lk /dx k =- 0, i e. the self-force /t =
The final solution of the problem of the electron and of the other
elementary particles can probably not be found on a classical basis.
is
quantum of action it may even be,
new fundamental constant of the dimension of a
Besides the introduction of Planck's
necessary to introduce a
length f But the above considerations show that, as long as one assumes
the existence of an energy -momentum tensor of the system, the theory of
relativity requires the vanishing of the self-force,
e of the
four-dimen-
sional divergence of this tensor
The fundamental equations of electrodynamics in stationary
matter
As shown by Loreiitz,f Maxwell's phenomenological equations of
72.
electrodynamics for stationary matter may be derived from the fundamental equations of the electron theory by avei aging over regions in
space which are small from the macroscopic point of view, but still so
number of electrons. Since the equations
of
election
the
(V 3, 0)
theory are covanant in for m, it must be possible
also to find the 'macroscopic equations of electrodynamics in moving
large that they contain a large
1
bodies by averaging oxer appropriate small space-time regions This
was actually done b\ Born and Dallenbach
However, if we assume the validity of Maxwell's phenomenological
|j
equations ior a body at rest, it is possible to find the corresponding equations in moving bodies simply by performing a Lorentz transformation
This method was used for the first time by Minkowski. j| The principle
of relativity requires ]bhat Maxwell's equations for stationary matter
must hold in that system of coordinates S Q in which the matter is at
rest, irrespective of
stars.
the velocity of this system with respect to the fixed
Thus we have
curl
in
E +~
0,
div
B :
curl
H -- D =
J/r,
div
(3D
Heisenbeig, Ann d Phys 32, 20 (1938)
Soe rof Chap I, p 21.
Ann 68, f>2G (1910), soo also A D Fokkor, Phil. Mag.
& Almkow ski Horn, Math
39', 404 (1920)
Dallonbach, DJSS Zuiich, 1918, Ann d Phij? 58,523(1919)
H
Gott Nachr p 53 (1908), Math Ann 68, 472 (1910)
ft Minkowski,
t
|
NON-TLOSKI) SYSTEMS
HHj
72
VII,
where E, D, H, B denote the electric field strength, electric displacement, magnetic field strength, and magnetic induction, respectively.
p and J are the macroscopic charge and current densities All these
quantities may in principle be determined by means of macroscopic
E and D, for instance, are defined as the forces on
experiments in *S
a small test body of unit charge inserted at the point considered into
small crevasses cut in the matter parallel or perpendicular to the field,
and B are the corresponding forces on a
respectively Similarly,
of unit
test
magnetic pole strength.
body
Besides the field equations (31) we have in isotropic dielectric and
paramagnetic substances the following constitutive equations connecting
the field variables with the constitution of the matter
magnetic
D
wheie
eE,
/xH
- aE,
(32)
the dielectric constant, ^ the magnetic permeability, and a the
The last equation is the mathematical expres-
is
electrical conductivity
Ohm's law
sion of
Minkowski's
equations in uniformly moving bodies
Consider two antisvmmetnral tensors Flk and Hlf
According to
F
the
tensor
defines
a
B
of
vectors
and E in the
(IV. 80, 80')
lk
pair
space
of
>S
coordinates
the
by
arbitrary system
equations
73.
field
is
an axial vector and
defines a polar vector
?E
(#>i, /'ai,/^)'
(//
a polar vector
axial vector
and an
*D
//,//,.,),
(^)
(*\H ^42^43)
Similarly, the tensor
lk
by the equations
(//,//,//)
(34)
Further, consider a four-vector with the components
in the
system
*S
When
(35)
(J/c,V>)
/,
f
the components of the tensors
Flk
7/, A
and J
one system of coordinates, we can calculate the components
are given
of these tensors in any other system by means of the transformation
equations (IV SI') and (IV 29) of antisymmetncal tensors and vectors.
For the components of
J we thus
L
get
(v
(36)
where
is
the velocity of
relative to S.
If
ELECTRODYNAMICS, THERMODYNAMICS
73
VII,
we now
H, B,
Flk
define the tensors
7/, A
197
so that the quantities E, D,
J, p are identical with the macroscopic electromagnetic variables
Q
of the matter, the field equaP in the rest system
E, D, H, B, J,
any system of coordinates must have the form
tions of electrodynamics in
(37 a)
For in the
'
<
system S
the equations (37) are then identical with
Maxwell's equations (31 ) and, since they are tensor equations, they must
hold in any system of meitia On account of (IV 187) the equations
(37 a)
may
rest
also
be wiitten
,,-n*
'*=-<),
(37 a')
^A
where F*^ is the pseudo-tensor dual to F, k
Using (33), (34), (35) in (37) we thus get
ordinates
p*
=- 0,
curlE-f-
curl
The quantities J -- ( J/r, ip)
densities in the system /S foi
,
^
i ]l
i
where u
is
87
co-
div
0,
(38 a)
(38 b)
be interpreted as current and charge
' 2//^
,_
(39)
^i'^k
an insulator, wliere J
--
in the rest system,
we
get fiom
(3(i)
the velocity of the ponderable matter relative to
Se
oi
from (376) we get the continuity equation
the equations
di\
may
-=
dt
Further,
system
f't
fD
e\ er\
in
-=/>8r
--=
/o8F,
<S*
Hence
(41)
the charge contained in an infinitesimal piece of matter of volume
2
z
8F^/(1 u /c ) is invariant and the current J is a puie conve(;tion
current
In the general case, the current density can always be written as a
of the convection current pu and the conduction current C
sum
J-pU + C,
(42)
NON CLOSED SYSTEMS
108
VII,
73
but this separation is not relativist ically invariant Even if p Q
in
the rest system, so that J
C is a pure conduction current in this
system, the charge density p in 8 will be different from zero, which
8. In fact,
means that we also have a convection current pu =
we
get from
(3ti)
in this case
(43)
We
can, however,
make
a relativistically invariant decomposition of
by writing
P QJJ
'
c,
where p
()
is
the (invariant) charge density
matter, and
U
is
the icst system
in
8Q
of the
*
,
I
9
the four- velocity of the ponderable inattei
s
l
(s,.s' 4 )
is
a four-
vector with the components
-
in
AS'
(JV.O)
(45)
Thus
H, B,
While,/, has a diiect physical meaning, the field variables E,
occurring in the field equations (38) have no simple physical significance
contrast to the field variables E, D, H, B in the rest system,
?
which could be detei mined by simple macroscopic experiments So far
they are defined only by the transformation equations by which they
may be expressed in terms of the field variables in the rest system 8.
Let us now consider the four-vector F defined by
t
From
(33)
we
V'"
Its
components of
get for the
components
E
(
"
in the rest
(^Ku^B)
aa
in
j(E.u)/r_
'
system are thus
F?
---
(E,0),
(49)
ELECTRODYNAMICS, THERMODYNAMICS
73
VII,
199
the four-force acting on a unit charge placed at rest relative to
the matter in a longitudinal crevasse cut in the matter If we put
is
-(uxB),
(50)
(48)
may
be written
and a comparison with (IV. 54) shows that
measured in the system S.
is
the force on the test body
in question
Similarly, the four- vector
.=
n jJ = / D_+(1 /l -u xH
(
.),
1^
U)/C
(52)
with the components
K ~
four-force on a unit test
Thus
(D,
0) in
the rest system, represents the
body mseited at rest in a transverse crevasse
also
D-D+^uxH)
(53)
has a simple physical meaning it represents the force on the test body
in question measured in the system 8.
.
Further,
(see
IV
if
F*k and H*k
108, 109),
F*
are the pseudo-tensors dual to
1
-
nu =
k
B-^-"f
and
lk
E)
l, j^.'D/f.
(B-)/c
K*
Flk
we can form the two pseudo-vectors
fi*
lk
c
(54)
uk ~
(55)
^* and
^Cf are obviously the four-forces acting on a unit magnetic pole
placed at rest relative to the matter in transversal and longitudinal
NON-CLOSED SYSTEMS
200
crevasses, respectively, as
is
seen at once
VII,
when one
73
considers the com-
ponents of these pseudo- vectors in the rest system. Thus
6=,B_ U **,
(56)
fl^H_
(57)
C
are the forces on these unit magnetic poles measured in the system 8.
The vectors
D, fi, B (or the four-vectors
v F*, K*) may thus
F K
t ,
by direct physical measurements performed
by an observer in $. By means of (50), (53), (56), and (57) we can now
also express Ftk and Hlk m terms of the quantities Fv K F*, K*. We get
in principle be obtained
(58)
where
8, A/m is
the Levi-
ivita
symbol defined
The equations
43
in
8 and,
hold
must
they
(58)
are easily seen to be true in the rest system
since both sides of the
equations transform like tensors,
Since the vector u is a constant
generally
curl(uxB)
curl(uxD)
Therefore the
field
curl
(ugrad)B-f udivB
</B
B -f
<^B
_-
(ugracl)B,
-
-(ugrad)D-fudivD
'
d*
0,
curl fi
at
divB
i
-=
(ugiad)D+pu
equations (38) can also be written in the form
where
we have
c
-- 0,
..
-(-(ugiad)B,
divD =
at
C/r,
(59)
(60)
p,
dD - dD
]XT.
+(ugrad)D
d(
.
B and D, and C is the conduction current defined by (42)
So far we have consideied only one material substance moving with
are the substantial time derivatives of
constant velocity u However, since the field equations are linear, the
fields are additive and the equations (37) must hold also in the case of
several bodies, separated
by a vacuum, moving uniformly with
different
The field equations (37) will, however, lepresent a good
approximation for a system of moving bodies only as long as the accelerations of the bodies due to the electromagnetic forces can be regarded as
velocities
small.
VII,
ELECTRODYNAMICS, THERMODYNAMICS
74
The constitutive equations
Boundary conditions
74.
According to the
201
in four-dimensional language.
equations of the set (32) the forces on a unit
first t\vo
and longitudinal crevasses are propoithe
and ju, according as the
constants
of
tional,
proportionality being
test body is an electric or a magnetic pole Therefore we must have
test
body
placed in transversal
8 =
- *,
K^eF^
T)
or
These equations
may
(6i)
jufi
Ft-f^Kf.
(62)
also be written
tf.^-^*^.
Ffk Uk = nH?,l\,
(63 a)
(63
ft)
the last equation being identical with the tensor equation
F.^+FulM.r^Mil'.+ HuUt + I/.il'J
(63r)
(61), (62) or (63) reduce to the first two
last equation of the set (32), Ohm's law, may be
In the rest system equations
equations of (32) The
put into the form
is
(64) aie
independent,
^^
seen from (45) and (49)
in the rest system Since s
as
(64)
when the vector equation
i
is
(64)
L\
-~ 0,
only the
(64)
first
is
written
down
three equations
equivalent to
"^
2
(6.-,)
v(i- AOhm's law can
)
On account
of (46) and (47)
J.
also be written
()
+ ^I'^^W*-
The field equations (37) together with the constitutive equations (63)
and (66) enable us to deteimme the field when the charge and current
distributions are
known
At the boundary between the ponderable matter and the vacuum the
and fl must be continuous, as is seen in
tangential components of
the usual way from (51)) by integrating these equations over infinitesimal
surfaces bounded by a small rectangle with two opposite sides immeIt is here
diately inside and outside the boundary of the matter
understood that the u occurring in the definition of and fl is put equal
to the velocity of the matter also outside the
boundary
Further,
we
by integration of (60) over a small cylinder with end surfaces
immediately outside and inside the boundary that the normal component
find
NOX-('L()SKI>
202
S\STKMS
VII,
74
B must
be continuous at the boundary, while the change AZ) n in the
normal component of D is equal to the surface density of charge on the
of
boundary
75.
Electromagnetic energy -momentum tensor and four-force
density
V we
In Chapter
is
theory
where
have seen that the four-force density in the electron
given by
f
Ji
"
AA, S A.'
density of this theory. This expression followed
irn mediately from the observation that by the very definition of the electric
s is the current
t
strength the force density in the rest system of the charge is yoE. In
ponderable matter with e and ^ different from 1 it is not so easy to find a
field
unique expression for the force density acting on the matter In the first
place, we have in general a conduction current in the rest system of the
matter, and even in an insulator
pE
it is
not evident that the force in the rest
strength is defined as the force on a
unit test body placed in a cre\ asse cut in the matter This uncei tainty in
the definition of the force density gives rise to a corresponding uncertainty
system
is
because the elect i ic
in the definition of the
However,
field
electromagnetic energy-momentum tensor.
F J which is the analogue
us consider the four-vector
let
ll
of the four-force density in the electron theory
(37)
we
From the field equations
get
T ,f'Ji
--
I1
f)///A
.]
FJ
Hence
tl
;>(^///A
-- M\i flii
nk
It
F-H
=-
-,
^A
S,*-
From
(33)
where
in the rest
matter.
and
(34)
iK
system
is
we get
,KJ)K
for the
^H
(67)
68 )
components of this tensor
/^-|(E D+H.B)
B llt
(69)
identical with Maxwell's stress tensor in ponderable
VII,
ELECTRODYNAMICS, THERMODYNAMICS
75
203
Further
(*V^42>^43) -= -S
I
S-r(ExH)
where
is
Poyntmg's
vector,
#44
and
-=-
h --
&,
KE.D + H
B).
(71)
In the rest system S and h are identical with the usually recognized
expressions for the electromagnetic energy flux and energy density in
stationary matter
(72)
where
=-.
(D x B)
J
The form of the equations (07) suggests that the left-hand side of (67)
the electromagnetic four-force density f and that *V tA as given by (68)
represents the electromagnetic energy-momentum tensor This would
is
mean that the
quantities S, h, g in e\ cry system of coordinates should be
as
the electromagnetic energy flux, energy density, and
mteipreted
momentum density, respectively The above expressions for S, /?, and
to Mmkowski,*)* in the case \\hen e -- ^
the corresponding expressions of the electron theory.
g are due
If we confine ourselves to
this
term
they reduce to
homogeneous and isotropic bodies it is
seen that the second term on the left-hand side of (67)
system
is
zero
easily
In the rest
is
on account of (32). Thus, if e and p. arc constant, this term is zero in S
but a vector with zero components in one system of coordinates is zero
in all systems Hence, inside a homogeneous and isotropic body we have
,
(73)
T
ie
=-/>E-f
(JxB) -p(E + -(uxB)\ (---(CxB))
C
C
(74)
-(E J)
= -E(pu+C)C
t See ref
195
-[f.ufE C]
NON-CLOSED SYSTEMS
204
75
VII,
In the rest system this expression for/4 is in accordance with Joule's
expression for the heat q developed in the body per unit time and volume.
In fact,
we have
'
'
c
Iii
of coordinates (f u) is the mechanical work, and
thus
must
C)
represent the heat production in accordance with
an arbitrary system
The
(IV 217)
four equations
represent in the usual
From
we
(73)
way
-*
/.
<
momentum
the
arid
76 )
energy laws
get
to the consideiations
50, we are here dealing with a typical
of
forces
a
which
change in the total proper mass of
example
pioduce
the matter
Accoidmg
Minkowski's
Ham
elect lomagnetiQ
identity
,V,,
energy-momentum tensor
as the tensor of the electron theory, but
S.A
In the rest
system
body on account
ha\ e
in
(FU HU )
-*;,//
satisfies
it is
the
(78)
not symmetrical,
/A,
(79)
the space part (09) is symmetric for an isotropic
but for the mixed space-time component we
iS'
of (32),
>S
#?4
-^c
*c((/?
- ^) =
i(ep,-
)(E
X H)
(80)
In any other system of leference we therefore also have*S
^ SKL even in
an isotropic body
This non-symmetry of Minkowski's energy-momentum tensor has
l/f
a long discussion in the literature | It was generally felt that
property represented a real difficulty for Minkowski's theory
Abraham J theiefore tried to construct a symmetrical expression for the
given
rise to
this
In the rest system $
electromagnetic energy-momentum tensor
Abraham's tensor agrees with (69), (70), and (71) at least for isotropic
t
M Abmham, Itend Pal 28 (1909) Ann d Phtj* 44,
)a), M \on Lauo, Die Kelatu'itaMheone, vol t,
(p
,
i-it
o])
HU9, \V
Pauli,
EmyU
Tfnimodynanncs, and
I'SXR, 1, 439 (1939)
|
\ol
11,
Math
("ostnoloyy,
II
?,s,s
54,
vol
2 (1920), p
Oxford,
1934,
337 (1914), \V Dalleiibach,
3rd ocl ^ 24, Biaunschxieig,
,
M7,R
Jg
C Tolman,
Tainni, Journ
Abraham, Rend Pal 28 (1909), Abraham Becker, Theorie der
6th od
Leipzig, 1933
Kelatinty,
oj
Phy*
Elcktnzitat,
VII,
ELECTRODYNAMICS, THERMODYNAMICS
75
bodies,
205
but instead of (72) Abraham assumed the electromagnetic
momentum
density to be given by
g= (ExH)^ c*
-
(81)
symmetrical in S
it is
any system S, but in any system other than S the
r
components of 8ff are not simply given by (69), (70), (71), and (81),
the expression of $^hl in terms of the field variables containing also
Since Abraham's tensor $^? ir
in the rest
system
symmetrical
is
the velocity u of the matter in a complicated way
br
/^ derived from this tensor by the equation
also deviates in general
from
(73) in a
The four-force density
*<A
complicated
way
In the rest
system we obviously have
fAbr
e in the rest
_ f + f/f_- 1 S
^
system Abraham 's force density
^
expression by the term
This term
__
y^
(83)
differs
from Minkowski's
yM>i
is
generally so small that
an experimental verification would be very difficult
Until quite recently, most physicists were inclined to adopt Abraham's
theory However, the question was never quite settled, and recently
Tammf has taken up the discussion again and he comes to the conclusion
that Minkowski's expression for the energy-momentum tensor is correct.
In the first place it should be remarked that, since an electromagnetic
field in ponderable matter is an essentially non-closed system, there is
no a priori reason for the symmetry of the energy-momentum tensor.
Abraham 's main argument for a symmetrical tensor was that the quantities of the macroscopic theory must be derivable from the corresponding
quantities in the electron theory by averaging over appropriate spacetime regions and, since the microscopic energy-momentum tensor s }k
is symmetrical, the averaged tensor lfr must also be symmetrical
But,
,v
remarked by Tarnm,|. the macroscopic tensor Slk is not simply the
average of s lk tilk must rather be defined so as to give the correct force
density and the correct moment of force, i e. we must have
as
'
t
|
(84)
dx k
dx k
Jg Tamm, see ref., p 204
Idem, private communication
NON-CLOSED SYSTEMS
206
and, on account of
(10),
>,,
From
tA
-,
_.
we can only conclude that
(84)
#i*
where
75
VII,
may
^~-KA>
possibly he a non-symmetrical tensor satisfying
**'*
0.
ar A
Fiom
(85)
we
see that
ti lk
will
be symmetrical only
l
and
this
is
Further,
_*,
'
^+
rA
if
^'
(86)7
V
not necessarily the case
Tamm could show that Abraham's expression in some special
wrong results, while Minkowski's expression for the
energy-momentum tensor is m accordance with the electron theory. It
must also be mentioned that Dallenbach| fiom perhaps not quite cogent
arguments lias given a general derivation of Mmkowski's tensor from
the electron theory
n the following section we shall meet another strong
in
favour
of
Minkow ski's theory
argument
cases leads to
While thus the electromagnetic energy-momentum tensor
sym metrical, we may assume
matter and
that the total
is
non-
energy-momentum tensor of
symmetric, since we then have to deal with a closed
This
however, that the mechanical energy -momentum
means,
system.
tensor of the matter must also be non-symmetrical This is not in contrafield is
diction with the considerations in
65, since
we there
considered a closed
mechanical system, and the expression (VI 66) for the momentum
2
density was derived from the explicit assumption g
S/c which must
be abandoned in our case if we adopt Minkowski's expression for the
,
electromagnetic
momentum
density
The propagation velocity
moving refractive body
76.
of the
energy of a light wave in a
In Chapters [ and 11 we have defined the direction and velocity of a
light ray in a transparent refractive body by means of Huyghens's
principle, and in 24 it was shown that the ray velocity so defined trans-
forms
like the velocity of a particle by Lorentz transformations, i e by
the equations (II 45-47
As a consequence of these equations we arrived
)
t See ref
p 195.
VII,
ELECTRODYNAMICS, THERMODYNAMICS
76
207
25 at the aberration formula (II 91 ) and at Fresnel 's formula (II 92)
which, as regards effects of the first order, are in agreement with the
in
experiments.
According to Maxwell's theory of light, optical phenomena in a rebody with the refractive index n are described by means of
fractive
Maxwell's phenomenological equations of electrodynamics for a substance with the electric and magnetic constants and connected with n
/u,
bv the equation
=->/M
87 )
where we can neglect all
a
Further,
transparent body which does not
phenomena
absorb any light must be regaided as a perfect insulator, i e we have
At
least this is true for sufficiently long waves,
dispersion
J -
-- 0,
0,
(88)
Now the ray velocity must be identical \vith the velocity with which
the energy in the wave is propagated In the aberration expenment, for
instance, the angle ot aboi ration is the angle through which the telescope
must be tilted in order to get the ray, e the eneigy, into the telescope
The direction of the ray velocity must therefore be the Scime as the direcWhen the energytion ot propagation of the; energy in the wave
momentum tensor of the electromagnetic' field is given, we can, however,
iind the velocity ot the energy by means of (VJ 9), e
i
u*
We must therefore require that
Sfh
(89)
wave transforms
as a particle velocity by Lorentz tiansformations. This means that the
quantity (VI 15) must be a four-vector As shown m 62, this is the
case only
if
the
u*
in the case of a light
energy-momentum tensor satisfies the condition (VI
19).
We shall now show that this condition is actually satisfied by Minkowski's
tensor (68)- (72), but not by Abraham's tensor, and this
argument in favour of Minkowski's theory
As mentioned
tion (VI
9) in
in
62, it is sufficient to
one system
is
a strong
prove the validity of the equa-
We choose to work in the rest system of the
We
can obviously confine ourselves to the consideration of a plane wave, for in the problems which were considered in
Chapters I and II the rrxlii of curvature of the wave fronts are large com-
refractive substance.
pared with the wa\ e-length (geometrical optics) and therefore the curved
wave fronts can at each place be approximated by plane waves,
As shown in Appendix 3, the most general solution of the field
0, representing a plane wave with the wave
equations with p -_ 0, J
NON-CLOSED SYSTEMS
208
normal n
(x.n)/ W
-.
" -
"
'
"
^-|x.n)At')
e(1
^
00)
,-
"
/
v/x
V/u,
Here e (1) and e
(2)
are two fixed unit vectors which are perpendicular to
each other and to the unit vector n,
(e
(1)
.e<
2
>)
(2
=- 0,
=- (e (1
>,n) -- (e >.n)
(91)
(x n)/w, and
--
(e^e,,)
g are arbitrary functions of the argument
w=
is
76
in the rest system, is
E _/(<"
/and
VII,
(92)
the phase velocity
Thus we get from Minkowski's expressions
S-r(ExH)^
where
e -= (e t
(72)
m the rest system
VM
X e2
(68)
f/ +f/ )e,
(93)
^- n,
(94)
the direction of propagation of the energy e coincides with the
direction of the wave normal n in this system Further, we have
hHence
i
u*
|(e
Further,
H/
(95)
'
_e
i/*e
--=
w,
e the velocity of the energy
system
^// )-/
From
we
(96)
we
get,
(S, ich)
is
(96)
equal to the phase velocity in the rest
by means of (VI
= (r f ?
)-
15
and
12),
e, ic
(98)
get from (69)
+f7> ^,
t
since
on account of
^
(91).
1)
d -f'^
1)
(
i
2>
4- g
^K
(99)
8 tK
Finally, (72) gives
*+!/
K.
(100)
VII,
ELECTRODYNAMICS, THERMODYNAMICS
76
200
For the quantity
we then
get,
by means of
and
(97), (99),
(100),
R lk
Thus the tensor
19) has the following
defined in (VI
components;
Hence the condition (VI 9) is satisfied and the velocity of propagation
of the energy u* is in every system of coordinates identical with the ray
velocity as determined by Huyghens's principle f
If v' is the velocity of the rest system S relative to a system of coordinates *S with the same orientation of the spatial axes as in A>, the
1
y/
transformation coefficients
with v
v',
e (lc in
(VI
maybe
120)
is
calculated from (VI. IS or
17'),
We
using (104).
u*
=-
--i
(
e+v'
is
in accordance
}
}
(106)
<J(JJ.)
which, of course,
given by (IV
Since the velocity of the energy in $'
this quantity
then get
17' or 18) are
with
(II.
55)
if
we put v
v'
and
u
{c/^(jji)}e in this latter formula and neglect terms of order higher
than the first in ?/. For the magnitude of u*' we get
=-
n
in
accordance with the Tresnel formula'
t See also
3595 60
Scheye,
(107)
Ann d Phyv
p
(II. 92).
(4),
30, 805 (1909)
NON-CLOSED SYSTEMS
210
76
VII,
On the other hand, if we adopt Abraham's expressions for the energymomentum tensor the equations (90)-(99) will still be valid in the rest
system, but instead of (100) we get from (81)
.
Hence,
A.
i
'
ff
-w^
for the quantity (101)
and the tensor K lk
four-vector and u*
To
O
^
we
fO
9\xv
<7
Jt
e ~- &
f\
O\
not be zero any more Thus l * will not be a
not transform like a paiticle velocity.
r
will
will
we again
use (104),
17'). Hence, with Abraham's expression for the energytensor we get for the velocity of the electromagnetic energy
and (VI
momentum
in the
/ 1
(108)
,,.
CL
get in this case
find explicit expressions for the \ector u*'
(105),
1\
i)
(&
>S"
system
e+v'-
(v '- e)c
+ (v' e)l-
Me,
(109)
which deviates from the ray velocity as defined by rluyghens's principle
by the last term Instead of (107) we now have
(109')
From
we
(109)
see that the direction of the energy flux in this case
is
different from the direction of the ray velocity as defined by Huyghens\s
principle, which would give rise to a change in the aberration formula
for light traversing a
being
even
of first
order
medium
in r'
of refractive index n
Further,
we
see
1,
the deviation
measure
very
moving transparent media
Unfortunately
first-order aberration effects in
difficult to
it is
fiom (109) that the velocity of the energy
from
differs
the phase velocity even if v' and e are parallel, in contrast to the ray
velocity which in this case is identical with the phase velocity This
strange result
is
connected with the following circumstance
While Mmkowski's four-force density/, is zero in the case considered,
Abraham's theory would give a non-vanishing force density on a homogeneous insulator
In the rest system
we have, according
>
dt
to (83),
(HO)
ELECTRODYNAMICS. THE KM OD\ N \M ITS
76
VII,
211
Thus, in this system, the electromagnetic energy is conserved, but this
From (104) and the transformation equations
will not be the case in /$"
1
f'Abr __ /Abr ~T-Ji
Ji
fAbr
we
ikJk
get
(111)
which is i/c times the mechanical work on the substance per unit time
and volume Thus, in *S' we have an exchange of energy between the
electromagnetic and the mechanical system, e a local absorption and
re-emission of light energy by the body This clearly shows that Minkowski's decomposition of the total energy-momentum tensor into an
electromagnetic and a mechanical part is more natural than Abraham's,
a transparent body being in M in ko \vski\s theory a system which does
i
not even locally exchange energy with the electromagnetic
The laws
77.
of
thermodynamics
As shown by Planckf and
may
Einstein;].,
in stationary
field.
matter
the usual laws of thermodynamics
the special theory of relathity
For
w
to
of
e
shall
confine
ourselves
the
consideration
simplicity
systems
consisting of a thermodynaimc Hind which can exert a normal pressure
be easily incoiporated
in
any sin face element In the rest system of the fluid the two laws
thermodynamics may be stated in the usual way
According to the first law the total energy of the system is a unique
function of its state In a thermodynamical process which gives rise to a
only on
of
change of
state, the
Q
change in energy d K
dE Q
--
is
given by
S^-fS/n
(112)
amount of heat transferred to the system by the process,
work done by the surroundings on the
In
a
reversible
system
process in which the volume V of the system
is increased by an infinitesimal amount, we have
where 8Q
while
8.4
is
the
is
the mechanical
-p Q dV \
(
(113)
where p is the pressure
Accoiding to the second law of thermodynamics, the entropy 8 in
the rest system is a function of the thermodynamical state The change
t
Planck, Berl fier p
116, 1391 (1907)
,
542 (1907),
Ann
d.
Fliyt
WienBer
}
Einstein, Jafnb f
Had nnd El 4,411
(1907)
76,
(1908),
Hasonohrl,
NON-CLOSED SYSTEMS
212
of
entropy content of a system by a small change of state
VII,
is
defined
77
by
where SC/^v is the amount of heat transferred to the system in a reversible
process winch brings about the change of state considered, and T is the
On the other hand, if the change of
an
about
irreversible
by
brought
process, we have always
absolute temperature of the system.
state
78.
is
Transformation properties
of the
thermodynamical quan-
tities
We
shall
now
establish the transformation properties of the thermo-
dynamical vai lables. Consider a system of inertia with respect to which
our thermodynamical system is moving with the constant velocity u.
Since the fluid in question is in equilibrium in the rest system, the total
momentum and energy are given by (VI 1 2, 1 13) which together with
1
(VI
and
100)
(II
34) lead to the equations
_ w va)
,-J/l --,r->
Jfl-</r,<l
'
(|ig)
Now
consider again a reversible process which brings about a change
In an arbitrary system of coordinates the
in the state of the system
first law of thermodynamics may then again be written
dK
8Q + &A,
(117)
where &Q and 8^4 are respectively the heat How through the boundary
and the work done by the surroundings on the system by the process
considered. This work is, however, not simply equal to
p dV, as it was
system. In order to preserve the equilibrium of the substance,
a constant velocity throughout the body, we must assume the existence of external forces besides the normal pressure The total extra force
in the body, which is necessary to keep up a constant u in *S\ is
in the rest
i.e
...
^=
at
c2
*(K+iM">),
\!(\~~u*/c
dt
(118)
from zero during the thermodynamical process considered, there must be an external force on the
and, since the right-hand side
is
different
VII,
ELECTRODYNAMICS, THERMODYNAMICS
78
213
system which carries out work of amount (F.u)dt ^ (u dG) during
the time dt. For the total work BA on the system we thus get
8^4
with
-^-/M/r+u.r/G,
From (1 17),
G given by the first equation (116),
(119)
(1 19),
and ( 116) we
thus get for constant u
SQ
=-=
dE +pdV-u.dG
^ dE
(u*/c*) d(
P ^)~(u*/c*)(d^
2
^!(i-~n
(dE-)rp
dV )^(lu
Q
/c
Hence, by means of (112) and
SQ
/c
=~-
(120)
).
(113),
SQ *J(l~u 2 /c
(121)
).
This transformation formula for the amount of heat transferred
through the boundary is identical with the formula (IV. 66) lor the nonmechanical energy produced inside a body, for instance in the case ot
the Joule heating effect
Further, we define the entropy and absolute temperature in any system
of coordinates by the equation
dS
= 8Q-.
(122)
Thus, for any reversible process without heat transfer, the entropy is
constant. Consider now a thermodynamical system in some internal
state originally at rest in a definite system of coordinates. If this system
accelerated reversibly and adiabatically to the velocity u without
is
any change in its internal state, the entropy must be constant during
the process on account of (122). Thus the entropy of the system is
independent of the velocity when the internal state is the same, which
means that the entropy must have the same value
coordinates
(cf.
the analogous consideration, p.
in
every system of
141),
,S'^,S">.
From
(122), (121),
and
(123)
we then
i.e.
is
an
(123)
get the following transformation
formula for the temperature
T-
T<V(l-~tt A
l2
)
For any irreversible change of state we have
ordinates
in every
124 )
system of co-
dS
>
6n
.
(125)
NON-CLOSED SYSTEMS
214
Four -dimensional formulation
dynamics
79.
of
the
VII,
79
laws of thermo-
was shown that the laws of conservation of energy and
momentum in differential form could be expressed by the tensor equation
In
69
it
PiT
-'
" 26)
This equation comprises the first law of thermodynamics. It is now
possible also to express the second law of thermodynamics in four-
dimensional language, f
Since the entropy is an additive quantity we can introduce the
s 8V is the entropy content of
entropy density s defined so that 8$
the volume element 8V. From (123) and the last equation (116) we get
at once the transformation equation
s
for the
time
8t
" 27)
entropy density. For the change of entropy in the infinitesimal
of the element considered we then have, according to (125),
>,
where 8Q
d(s
is
8V) /dt is
(128)
the heat transferred to the element in the time
the substantial time derivative and, thus,
we
8t
Here,
get by means
of (IV. 200 and 204)
=
g+(grad)
SU
(div(
We
can
now
S+S divu
(129)
~)sr.
introduce the four-current density of entropy
(130)
by analogy with the four-current density of electric charge (V 3) On
account of (127) and (IV. 39) the four- vector /S, may also be written
(131)
by analogy with
f
C.
Tolman,
(V. 9).
Rclatn'ity, Thcrmotli/ttamiif
and Cosmology,
71,
Oxford 1934.
VII,
ELECTRODYNAMICS, THERMODYNAMICS
79
By means
of (129) and (130) the left-hand side of (128) takes the form
'SKSJ
-*8S,
8r8,r 4 /i
Si]
is
(132)
3x l
d,r t
where
215
the four-dimensional volume element defined by
(IV. 121)
Since the right-hand side of ( 128) is an invariant on account of (121)
(124), the second law of thermodynamics may be written in the
and
simple invariant form
,,
*>
^ nQ
(133 >
monatomic gases
For a monatomic gas consisting of N molecules we have in the rest
system the usual expressions for the energy, entropy, etc., at least for
moderate temperatures, where the mean kinetic energy of the molecules
2
is small compared with the rest
energy m c of a molecule. Thus if
80. Ideal
m
we have
in the rest
where k
is
(134)
1,
system the equation of state
(135)
Boltzmann's constant. Further,
#0
- Nn^+lNkT*,
where the energy has been normalized so that
EQ is equal
(136)
to the
sum
of
the rest energies of the molecules at zero temperature.
From (114), (135), and (136) we get in the usual way
(137)
and by integration
|A Hn T"+Nkln F
T
+ C,
(138)
C is a constant which is independent of T and F.
In a system of coordinates in which the gas has the constant macroscopic velocity u, we get by means of (116), (124), and (135)
where
e.
(139)
P V = NkT,
the equation of state of an ideal gas is invariant in form. Further,
O ___
M
(140)
fi
*NklnT+NklnV-tNlslnJ(l-'U*lc*) + C
NON-CLOSED SYSTEMS
216
The energy-momentum tensor
is
80
VII,
given by (VI. 104)
(141)
= N/V is the number of molecules per unit volume in the rest
Thus
we have the following relation between the pressure p,
system.
the rest density //,, and the temperature T Q
where n
v?kT
(142)^
(
-^~+pl"/c'-
Black -body radiation
The electromagnetic radiation
81.
inside a hollow enclosure in equilibrium
with the walls at a definite temperature can be treated as a perfect fluid.
In the rest system of the walls the flux of electromagnetic radiation is
zero at every point and, according to 8tefan-Boltzmann 's law, the energy
density A
is
given by the formula
where
is
7-6237
10~ 15 erg cm.- 3 deg.~ 4
Stefan-Boltzmann 'a constant. Thus h
of the enclosure.
The radiation
P
=a
is
(144)
independent of the volume
p which is
exerts a normal pressure
l^
= -= Mi/r1
7/
-
<
I45 >
For the total energy of the radiation we now get
(146)
and, on account of (114),
iV Q
By
integration
we
get
*aFT
*
.
(147)
(148)
where the constant of integration has been chosen so as to make 8
for F -=
and T = 0.
In the system of coordinates, where the rest system is moving with
velocity u, we then get, on account of (116), (123), (124),
""
G-
.Q
_
""
4
*
/7F773
QV
(149)
VII,
ELECTRODYNAMICS, THERMODYNAMICS
81
The
relation
symmetric
between G, E,
electrostatic
system
and u
is
217
the same as for a spherically
(cf. (23)).
The energy-momentum tensor
for
black-body radiation
is
**
again
(
150 >
with the relation (145) between the pressure and the density of the
It satisfies the same relation
rest mass.
energy-momentum tensor of an arbitrary electroV.
108). This should also be expected, since the macromagnetic
scopic energy-momentum tensor of the black-body radiation is the
as the electromagnetic
field (cf.
average of the electromagnetic energy-momentum tensor in
the canonical ensemble corresponding to the temperature T
statistical
'.
VIII
THE FOUNDATIONS OF THE GENERAL
THEORY OF RELATIVITY
82,
The general
principle of relativity
ACCORDING
to the special principle of relativity winch is the basis of the
special theory of relativity, all systems of inertia, i.e. all rigid systems of
reference moving with constant velocity relath e to the fixed stars, are
completely equivalent as regards our description of nature. Mathematically this principle found its expression in the covariance of the
fundamental equations of physics under Lorentz transformations In
spite of the inner consistency
theory of relativity,
it is,
and harmony which characterize the special
however, extremely unsatisfactory that this
theory also distinguishes certain systems of reference, the systems of
inertia, from all other conceivable systems of reference. This defect was
felt especially serious in the treatment of the so-called clock paiadox
mentioned in 20 There we actually had to refrain from giving a real
solution of the clock paradox The question was simply rejected by a
reference to the fact that the system of coordinates $*, following the
moving
clock,
is
a system of inertia for only part of the time and that a
discussion of the problem in this system of coordinates therefore falls
beyond the scope of the special theory of relativity.
However,
it
seems to be rather
difficult
beforehand to acknowledge
the accelerated systems of reference as equivalent to the systems of
phenomena (When, in the
inertia as regards the description of natural
following chapters,
always have
in
we speak
of an accelerated system, simply, we
relative to the systems of
mind a system accelerated
For example, if we consider a purely
)
mechanical system consisting of a number of material particles acted
upon by given forces, and with velocities small compared with the
velocity of light relative to a system of inertia, Newton's fundamental
equations of mechanics may be applied with good approximation in the
inertia or to the fixed stars
description of this system.
On
the other hand,
if
we wish
to describe the
system in an accelerated system of reference, we must introduce, as is
well known, so-called fictitious forces (centrifugal forces, Coriolis forces,
etc.) which have no connexion whatever with the physical properties of
the mechanical system itself. In fact, they depend exclusively on the
acceleration of the system of reference introduced relative to the systems
of inertia.
82
VIII,
It
was
THE GENERAL THEORY OF RELATIVITY
just for this reason that
219
Newton introduced the concept
of
absolute space which should represent the system of reference where
the laws of nature assume the simplest and most natural form. However,
as mentioned at the beginning of Chapter II, the notion of absolute space
meaning as soon as the special principle of relativity was
lost its physical
generally accepted, for as a consequence of this principle it became
impossible by any experiment to decide which system of inertia had to
be regarded as the absolute system
Therefore, Einsteinf advocated
of
the
fictitious
forces
in accelerated systems of
interpretation
reference instead of regarding them as an expression of a difference
in principle between the fundamental equations in uniformly moving
a
new
and accelerated systems he considered both kinds of systems of reference to be completely equivalent as regards the form of the fundamental equations and the 'fictitious forces were treated as real forces on
'
the same footing as any other force of nature. The reason for the occurrence in accelerated systems of reference of such peculiar forces should,
according to this new idea, be sought in the circumstance that the distant
masses of the fixed stars are accelerated relative to these systems of
reference. The 'fictitious forces' are thus treated as a kind of gravitational force, the acceleration of the distant masses causing a 'field of
gravitation' in the system of reference considered.
The idea that the
acceleration of the distant masses can produce a
gravitational field which is not perceptible in a system of inertia is not
more artificial than, for example, the fact that an electrostatic system
has zero magnetic
field in the rest system of inertia of the charges, while
a magnetic field is present in every system of inertia in which the charges
are moving with constant velocity. The cause of the appearance of a
magnetic field in the 'moving' system of inertia must be sought in the
motion of the electric charges relative to these systems, and the appearance of the magnetic field can at any rate not be taken as an indication
that the fundamental equations of electromagnetics have different
forms in different systems of inertia The only essential difference
between the two cases considered
field can be found
the magnetic
is
the circumstance that the cause of
in the state of
motion of
terrestrial
systems (viz. that of the charges), while the origin of the gravitational
fields in accelerated systems must be sought in the state of motion of the
distant celestial masses.
Previously the effect of the celestial masses
negligible now, however, we must include
had been considered to be
t
ibid.
Einstein, Jahrb f.
Rad und EL
38, 355, 443 (1912), Phys
ZS
4, 411 (1907),
14, 1249 (1913)
Ann
d.
Phys 35, 898 (1911),
THE FOUNDATIONS OF
220
VIII,
the distant masses in the physical system considered. Only
work in special systems of reference, viz. systems of inertia,
82
when we
is it
not
necessary to include the distant masses in our considerations, and this is
the only point which distinguishes the systems of inertia from other
systems of reference.
It can, however, be
assumed
that all systems of refer-
ence are equivalent with respect to the formulation of the fundamental laws
of physics. This is the so-called general principle of relativity.
The
83.
principle of equivalence
The interpretation of the
'fictitious forces' as gravitational forces is
corroborated decisively by the fact that they have an essential property
in common with the usual gravitational fields, viz. the property to give
all free particles
the same acceleration irrespective of the mass of the
'fictitious forces' have this
the
first
to
this
and
was
Galileo
property,
prove
property for the gravitational field ol the earth As a result of his experiments he was able to
particles
It is
immediately clear that the
make the statement that all bodies 'are falling with equal speed' in empty
space. This result simply expresses the fact that the force with which the
gravitational field of the earth affects a particle is proportional to the
inertial
mass of the
particle
which determines the inertia of the particle
against changes of motion As long as the velocity of the particle is
small relative to the velocity of light, its motion in the direction of the
gravitational field
is
therefore given
mx
by the equation
wg,
where m is the mass of the particle and x is the acceleration of the particle
The quantity g is a measure
in the direction of the gravitational field
of the strength of the gravitational field
and
is
independent of the mass
of the particle This circumstance is frequently expressed by stating
that the ratio between the inertial mass of a particle and its gravitational
a universal constant, depending only on the units in which the
quantities in question are measured. This theorem has now been proved
mass
is
by a large number of experiments,! the most accurate of which are those
performed by Eotvos and by Zeeman. The ratio of the inertial and
gravitational mass was always found to be the same. A particular
was attached to the experiments of Southerns and Zeeman
with uranium, which at that time was known to have a great mass
interest
v Eotvos, Math u nalurw Ber aua Ungarn, 8, 65 (1890), Ann. d Phys. 59,
Southerns, I' roc ftoi/ tfor London, A, 84, 325 (1910); P. Zeeman, Proc.
Amst 20, 542 (1917) H. v Eotvos, D. PekAr, and E. Fekote, Ann. d Phys 68, 11
f
354
181)6), L.
(1922).
VIII,
THE GENERAL THEORY OF RELATIVITY
83
221
In Chapter III we have seen that any energy of amount E
2
E/c a theorem which has been
corresponds to an inertial mass m
defect.
by numerous nuclear transformation processes
The
mass
which
is determined in a mass spectrograph obviously
32).
(cf.
is the inertial mass, and Zeeman's result now shows that the binding
energy of the uranium nucleus which appears in the mass defect also
corresponds to a gravitational mass which has the same universal ratio
to the inertial mass as for all other types of mass.
verified experimentally
In view of the property just discussed, a gravitational field may thus
be characterized by the 'gravitational acceleration independent of the
mass of the test particle, and this applies both to the usual gravitational
'
fields due, for
example, to the gravitation of the earth or the sun, and
to those gravitational fields which appear in accelerated systems of
reference and which are due to the distant masses of fixed stars Actually
the gravitational field on the surface of the rotating earth is a mixture
of these two types of field, the centrifugal force due to the rotation of
the earth being in general* not negligible compared with the force due
to the attraction of the test body by the mass of the earth. It is thus
quite natural to assume that both types of gravitational fields are of the
same nature and obey the same fundamental laws. This assumption is
often referred to as the principle of equivalence. It is true that the
gravitational fields due to the distant masses can be made to disappear
by a suitable choice of the system of reference,
of inertia as
from
'close'
viz. by choosing a system
system of reference, while the gravitational fields arising
masses such as that of the earth or the sun cannot be
'transformed away' by a proper choice of the system of reference; the
latter fields will therefore be referred to as permanent gravitational fields.
In this respect, however, the situation is quite similar to the case of
the magnetic fields with which the gravitational fields were compared
in
82. In some cases, viz. when the charges producing the electro-
magnetic field have the same constant velocity relative to the fixed stars,
it is possible completely to transform away the
magnetic field by choosing
the rest system of the charges as system of reference, for in this system
the field will be a purely electrostatic field. In general it will, however,
not be possible to choose a system of reference in which the magnetic
disappears everywhere in this system. Nevertheless, in this case
the electromagnetic field is not considered as essentially different from
the field of a system where the magnetic field can be transformed away.
field
In
all
cases the electromagnetic field obeys the
tions, viz.
Maxwell's equations.
same fundamental equa-
THE FOUNDATIONS OF
222
VIII,
83
The most important task will now be to look for the general fundamental laws which all types of gravitational fields must obey. In the
first place, we must, however, try to find the field functions which can
give an adequate description of the fields of gravitation. To this end we
shall first consider the simple case where no permanent gravitational
fields are present. By a suitable choice of the system of reference, viz.
in a system of inertia, the field of gravitation will disappear and we can
apply the laws of the special theory in this system A simple transforma-
tion to accelerated systems of reference permits the determination of the
field quantities describing the gravitational fields in accelerated systems,
and, according to the principle of equivalence, these quantities may be
assumed to give a correct description also of the more general permanent
fields of gravitation.
84,
Uniformly rotating systems
of coordinates.
Space and time
in the general theory of relativity
The development of the ideas underlying the general principle of
relativity now leads, as we shall see, to a still more radical revision of our
conceptions of space and time than that required by the special principle
of relativity To illustrate the character of the problems with which we
.
we
with the consideration of a very simple
accelerated system of coordinates, viz. a rigid uniformly rotating system.
Let / denote a certain system of inertia in a part of space so far from all
are confronted,
masses that
Z,
all
shall start
gravitational effects can be neglected; further, let X,
',
T denote the usual space-time coordinates defined in the way discussed
we can obviously just
curvilinear
for
the fixation of the
coordinates
general
employ
If
in
we
confine
to
the
consideration of
ourselves
points
physical space.
events in the ^YF-plane we can, for instance, introduce polar coordinates
in
16
and
17.
Instead of Cartesian coordinates
as well
(R,9) by means of the equations
-^
Rcose,
y=-j?sin0.
Now we
can define a uniformly rotating system of coordinates
with spatial coordinates
rcos#,
y =- rsin#
by means of the transformation equations
r = R,
# = 6-ajT.
(i)
S
(2)
(3)
Any fixed point in the rotating system corresponding to constant values
of (x,y) or (r,$) obviously performs a circular motion relative to / with
the constant angular velocity
ordinates coincide.
cu.
For
the two systems of co-
THE GENERAL THEORY OF RELATIVITY
84
VIII,
Thus
223
for all points with
r
-R<
c/aj
(4)
the rotating system of reference may be represented by a uniformly
rotating material disk. Each point p on this disk is characterized by a
pair of numbers (x,y) or (r,&) which are equal to the coordinates (X, Y)
or (R,B) of that point in the fixed JCF-plaiie with which the point p
coincides at the moment when the clocks in the system of inertia /
record the time T
0.
For the measurement of distances between fixed points on the rotating
disk we shall use standard measuring-rods of the same kind as those used
in the systems of inertia, but now at rest relative to the rotating disk.
In connexion with the process of measuring distances in accelerated
systems of reference a problem arises which did not occur in inertial
systems. If the measuring-rods in one way or other are kept in a fixed
position relative to an accelerated system of reference, they will generally
be submitted to forces which may cause a deformation of the measuringrods for, according to the special theory of relativity, no absolutely
rigid bodies can exist, since they would provide a means of transmitting signals with velocities larger than c.
Consider, for example, a measuring-rod, one end of which is attached
to the point (/%#) on the rotating disk and which lies in the direction
;
of the radius
The
centrifugal forces will then undoubtedly cause a
This deformation will, however,
lengthening of the measuring-rod
depend on the elastic properties of the material from which the
measuring-rods are made, and all such deformations of the measuringrod can therefore easily be corrected for. Now we make the assumption
that measuring-rods on the disk, after insertion of these corrections, have
same length relative to I as the standard measuring-rod in the
exactly the
system of inertia 7 which at the moment considered has the same velocity
as the measuring -rod on the rotating disk. In general, we shall assume
that the (corrected) standard measuring -rods in an accelerated system relative
to the
measuring-rods in I are subjected
means
to
Lorentz contractions only, which
that the lengths of the rods are independent of the accelerations
relative to 7.
we
two points (r,#) and
(r+dr,#) on the disk with a standard measuring-rod on the disk we get
If
therefore measure the distance between
the value
-,
dcr
-,
dr;
,-\
(5)
for the velocity of the measuring-rod relative to 7 is perpendicular to
the direction of the rod and thus does not give rise to any Lorentz
THE FOUNDATIONS OF
224
On
VIII,
84
we
consider two points with the coordinates (r,&) and (r,&+d&) on the disk, a measuring-rod connecting
these two points will have the velocity ra> relative to / in its own direction
contraction.
and
it
the other hand,
will therefore
if
be contracted relative to the measuring-rods in /,
Hence the distance
in accordance with Lorentz's formula (II. 33).
between the two points measured with the contracted measuring-rod
For the distance da between two neighbouring points (r,#) and
(r+dr,#-f d#), measured with a standard measuring-rod on the disk,
we
get similarly
This follows immediately by means of the Pythagorean theorem if we
consider the measuring procedure from the point of view of an observer
in the system of inertia /, taking into account the fact that the measuring-
rod moving with the rotating disk relative to the measuring-rods in /
is contracted in the direction of the velocity, but unchanged in a direction
perpendicular to the direction of motion.
It is now immediately clear that the geometrical theorems obtained
by means of measurements with standard measuring-rods at rest relative
to the disk in general will deviate from the theorems of Euclidian
geometry. Consider, for example, the curve given by the equation
constant.
(5) this curve will represent a circle with radius r. The
of
this
circle will, however, according to (6), have the length
periphery
According to
27T
rd&
o
Therefore the ratio of the length of the periphery to the radius will not
be 27T, but
2
Consequently we see that the recognition of the general principle of
relativity, according to
which the accelerated systems of coordinates
are equivalent to the systems of inertia for the description of nature,
some cases to abandon the Euclidean geometry which, as
forces us in
particularly advocated
tivity, was regarded as
by Kant and even in the special theory of relaan indispensable foundation of all description of
space. This also has the consequence
(cf.
87) that
it is
not possible in
THE
84
VIII,
CSKNT ERAL
THEORY OF RELATIVITY
225
general to use Cartesian space coordinates in accelerated systems of
reference and that we have to make use of general curvilinear coordinates
for the specification of the points in
physica4 space.
Similarly, the general principle of relativity also requires a
renewed
revision of the notion of time.
In the special theory of relativity the
time in a system of inertia was simply defined by means of standard
clocks placed at different points in the system and regulated by means
of light signals
the way described in 16. Two clocks which had been
synchronized in this way remained synchronous. Likewise we could now
think of defining the time in the rotating system of coordinates by means
of standard clocks inserted at rest everywhere in this system and set
according to the standard clocks in the system of inertia /, for example,
by putting them to zero at the moments when the clocks in / with which
they coincide show zero. Then we have for the time t thus defined in the
for T
0. A standard clock at the point r, # on
rotating system t =
the disk has, however, a velocity TOJ relative to /, and therefore it will
be retarded relative to the clocks in / in agreement with equation
e at a later time we have
(II. 36),
TV(l-r
o; /c
(9)
).
In accordance with the assumption made in
the velocity, not the accelerations relative to
20,
it is
implied that only
the rate of
/, will influence
Actually, sufficiently strong accelerations will of
course more or less influence the rate of a real clock (cf. a watch which is
a standard clock
dropped on the floor), but such an effect which depends on the material of
which the clock is made can be corrected for, just as was the case
with the measuring-rods.
The description of time in the rotating system which we obtain by
using the time variable t defined by (9) is, however, although admissible
in principle, highly unpractical. Imagine, for instance, a light source
placed at the point A with coordinates (r, #) and which
The number of waves emitted
emits light with the proper frequency v
1 is then
to t
in the time-interval from t
by definition equal to v
(an atom) which
is
1 is
to T
The number of waves emitted during the time from T
2
2
2
of
waves
number
The
same
to
v
r
co
therefore, according
(l
(9),
/c )*.
will also arrive at the centre
T=
=
to
to
T=
or, since
during the time-interval from
for r
0, in the time-interval from
0)
1.
The number of
light
waves which are emitted from the point A per
t is thus
larger than the number of waves
unit time in the time-scale
3595.60
(r
=T
THE FOUNDATIONS OF
226
84
VIII,
per unit time, and with this time variable we get
a very complicated description of the propagation of light. This consideration shows that in general it is not convenient in accelerated
arriving at the centre
systems of reference to use a time variable defined by standard clocks,
and that a much simpler description may be obtained when one uses
clocks of a different rate. In the case of the rotating disk, for example,
will be most convenient to use coordinate clocks whose rate at any place
it
is (1
r 2o> 2 /c 2 )~*
times faster than the rate of the corresponding standard
t defined
by these co-
clock, for this
means that the time parameter
ordinate clocks
is
formation
t
instead of
T in
identical with the time
=-
7, i.e.
we have
the trans-
(10)
(9).
In principle
it is
admissible, however, to use coordinate clocks of an
arbitrary rate, provided that the time variable t defined by these coordinate clocks gives a reasonable chronological ordering of the physical
events. In accelerated systems of reference the spatial and temporal
coordinates thus lose every physical significance, they simply represent
a certain arbitrary, but unambiguous, numbering of the physical events.
Non- Euclidean geometry. The metric tensor
As we have seen, the spatial geometry on the rotating
85.
disk
is
non-
geometrical experience in the three-dimensional
in complete agreement with the theorems of Euclidean
Euclidean. Although
all
physical space is
geometry, the notion of non- Euclidean geometry in two dimensions is
by no means foreign to us, since we meet examples, of such geometries
on every curved surface A well-known example is the spherical geometry
on spherical surfaces. As an introduction to the non-Euclidean geometries in w-dimensional space,
we shall therefore
consider the geometry
in a three-dimen-
on an arbitrary two-dimensional surface embedded
sional Euclidean space. If x, y> z are Cartesian coordinates in this space,
the surface in question may be given by a parametric representation
where F, G
- F(x\x^
* - H(x\x*)>
y - 6V,*
H are given functions of the two parameters x
certain intervals.
(11)
),
differentiation of (11)
By
,
dx
dF
'dF
-ax
,
dx l -\
dx
we
and x 2
in
obtain
^dx*
(11')
dz
&H
1
dH
--
dx l -\
^
.
VIII,
THE GENERAL THEORY OF RELATIVITY
85
227
The distance ds between two adjacent points on the surface corresponding
and (x l +dx l ,x 2 +dx 2 ), respectively, is
to the parameter values (x l ,x 2 )
iven b ^
ds 2
dx 2 +dy 2 +dz 2
2
l
dx, dy, dz are linear expressions in dx and dx given by (11').
2
Using these expressions we therefore get ds expressed as a homo1
2
geneous quadratic form in dx and dx i.e.
where
ds
=g
(dx^+g ia (dai da*)+gn (dxVx l )+g^dx^
l
ll
(12)
with
3F 3F
dG 8G
dH dH
(12
The angle between two line elements corresponding to the increments
l
l
2
and (A# = (Ax 1 A.r 2 ), respectively, for the para(S#
(Sx ,$x
7
meters
(x ,x
2
)
is
given by the equation
(13)
SsAs
where (So:, Sz/, 8z) and (A#, A?/, Az) are the increments in x, y, z obtained
from (IT) by replacing dx = (dx l ,dx 2 by (8x ) and (Aa: 1 ), respectively,
85 and As being the lengths of the line elements. Applying (11'), (12),
and (12') this equation can be written in the form
1
cos
The
line
i*
l1
elements (8x l )
2i
(Sx ,Sx
2
)
,8
and (As 1
(Ao;
Ao: 2 ) also define
an infinitesimal parallelogram on the surface with the area
85 As sinfl,
da
where
(14')
given by (14).
The curves on the surface which are obtained from
is
(11)
by putting
constant,
(15 a)
constant,
(156)
respectively, are called coordinate curves. Every point on the surface
is the point of intersection of two coordinate curves from the manifolds
1
(15a) and (156), respectively. If the line elements (8^') and (A# ') are
lying in the directions of these coordinate curves we have
(&**)
(cfeSO)
and
(A**)
THE FOUNDATIONS OF
228
and we get from
(12)
and
^=
VIII,
85
(14)
ten)*
^>
AS
(022)*
dx*
7--77,
(011022)*
where
011
012
021
022
the determinant corresponding to the scheme of numbers g lk
area du of the parallelogram we thus get from (14')
is
da=
For the
l
<Igdx dx*.
(17)
If the parametric representation (11) of the surface has the property
l
2
that every set of values of the parameters x
(x ,x ) corresponds to
2
1
one and only one point on the surface, x
(.r ,^ ) represents a set of
l
general curvilinear coordinates (Gaussian coordinates) on the twoAll the fundamental geometrical quantities can
then be expressed in terms of these coordinates alone without reference
dimensional surface.
to the variables of the three-dimensional space in which the surface was
supposed to be embedded. lfy tk ~- <J lk (x ) are given functions of the col
ordinates
(r'),
the line element
(Is
summation over
The angle
by (14)
2 _-=
line
(18)
and
2 being implied in this
elements ($x
___^
ox LX k
o,'Sx^x
y
tA
.
cosS 6
(12),
K
g lk dx*dx
for the values
between the two
'
given by
-*
and k
expression.
is
and
(Ao;
is
given
,
and the area of the parallelogram defined by two line elements in the
directions of the coordinate curves is given by (17). The quantities g lk
the components of the so-called metric tensor thus determine the
geometry on the surface which in general will be non-Euclidean.
86.
Geodesic lines
The
straight lines which in Euclidean
geometry may be defined as the
curves of shortest distance are in the more general case replaced by the
geodesic lines which likewise may be defined by a variational principle.
Let us consider an arbitrary curve connecting two points P and P2 on
the two-dimensional surface. In a parametric representation of this
THE GENERAL THEORY OF RELATIVITY
86
VIII,
curve the Gaussian coordinates x l
Or
,:*:
may
229
be regarded as certain
functions of an arbitrary parameter A in the interval
A!
xl
i.e.
Let
L(x'
(20)
a^A)
A2
(i
(20)
1,2).
x l ) be a given function of the variables
and
xl
The curve
<A<
dx l /d\.
which gives the integral
A2
f
L(x ,&)d\
A\
a stationary value for all infinitesimal variations of the curve connecting
the fixed points Pl and P2 is then determined by the condition
r*
L(x\x')d\
----=
(21)
At
for all variations 8^ (A) satisfying the
boundary condition
So^Aj) ==
Now we
So; (A 2 )
0.
(22)
have
A2
A2
L dX
Jg
8^(A)
+ g 8^(A)
^/A,
and since 8x = d(8x )/dA, we obtain by partial integration of the
term in (23), taking into account the boundary condition (22),
l
(23)
AI
AI
l
last
This integral can only be zero for all imaginable variations Saf(A) if the
factor in the square brackets is zero along the whole curve. The variational principle (21), (22)
is
thus equivalent to the Euler differential
equations
(24)
(
;
If
have
is
a homogeneous function of the variables x l of nth degree,
we
THE FOUNDATIONS OF
230
and for ft
1 it is
(25)
we
dL
must be constant
equations (24). From (24) and
easily seen that the function
along the curve defined by the differential
86
VIII,
l
L(x x
get
___
dL
3L
.,
/yl
/yt
___
d ldL\
I
.,
>>*
,
I
dL dxl
d IdL
__
fy't
\
I
tn
dL
(26)
Hence
If
-=f=-
1)^ = 0.
(n
1
we can thus conclude
L(x\x
is
(27)
d\
an integral of the equations
that
l
)
constant
(28)
(24).
Now we define the geodesic lines by the variational principle (21) with
(29)
The corresponding Euler equations
dx*\
d'l
(24) are
1
dda* dx>
(
'
which represent two differential equations of second order for the two
l
functions # l'(A). Since Lin (29) is homogeneous of the second degree in x
,
-=
constant
and by a suitable choice of the parameter A we can
always choose the constant on the right-hand side of (31) to be equal to
1 This
obviously means that we choose the length s of the curve measured
the
geodesic line itself as parameter. (30) and (31) then take the
along
form
is
an integral of
(30),
dl
ds
It
is
now
_
-
gtk
\
"d)
ki
~8x>
~fate'
gik
Us ds
'
seen at once that the curves defined by (30) and (31) also
satisfy the Euler equations (24) with
.
which means that the geodesic
^ *r
lines also satisfy the variational
equation
the distance between two points measured along the geodesic line
connecting two arbitrary points has a stationary value.
i.e.
VIII,
THE GENERAL THEORY OF RELATIVITY
86
231
Geodesic lines, angles between two intersecting geodesic lines as well
as the distance between two points measured along the connecting
geodesic line, are now completely determined by (32), (19), and (18) when
the metric tensor g lk is given as a function of the general coordinates (x l ).
Then we can also deduce geometrical theorems regarding triangles formed
by geodesic
lines, etc.,
dimensional space
is
on the surface,
completely defined
i.e.
the geometry of the two-
l
by the quantities g lk (x ).
Determination of the metric tensor by direct measurements.
Geometry in n -dimensional space
The preceding deductions are, of course, completely independent of the
87.
system of coordinates employed If we use another set of curvilinear
connected with the original coordinates x by a trans-
coordinates x'
formation
we have
dx
f
'
c\r
i.e.
r/A
(35)
the differentials of the coordinates are connected by linear equations.
2
if we eliminate the dx< by means of (35) in (IS), ds will also be
Therefore
a homogeneous quadratic form in dx'
ds*
where the
coefficients
ordinates x'
=g
g\ k
dx dx k
l
lk
~-
dx' dx' k
l
g\ k
(36)
can be regarded as functions of the
new
co-
Since the geodesic lines are defined by the invariant variational principle (21), (29), it is obvious that the differential equations for the
geodesic lines expressed in the
new coordinates are obtained from
(30) or
by simply replacing g lk and x by g\ k and x'\ respectively, in other
words, the equations -(30) are covariant or form-invariant. The same
l
(32)
holds for the equation (19).
If it is possible by a transformation of the type (34) to introduce
k
coordinates
f (x ) such that the line element in the new coordinates
assumes the form
ds 2
(dX
l
)
+(dX*)
= S dX dX k
l
lk
(37)
the geometry on the surface is called Euclidean. In this case the co1
play the same part as do Cartesian coordinates in a
ordinates
Euclidean plane. The differential equations (32) for the geodesic lines
reduce in these coordinates to the equations
THE FOUNDATIONS OF
232
87
VIII,
which have the same form as the equations
for straight lines in Cartesian
surfaces
are
of
such
cylinders and cones which
Examples
can be unfolded on a plane without internal deformation. All geometrical
coordinates.
theorems about triangles or other figures on such surfaces are identical
with the theorems of Euclidean geometry, and if we are interested only
in the two-dimensional
geometry on the surface,
all
such surfaces
may be
regarded as identical.
In general it is not possible to introduce on the surface such coordinates (Cartesian coordinates) for which the line element assumes the form
(37). In this case the geometry on the surface is a non-Euclidean general
Riemanman geometry. In any
case it is possible by means of measurements on the surface to determine the geometry on the surface without
referring to the three-dimensional Euclidean space m which the surface
is embedded.
Let us assume that we have introduced an arbitrary
of
coordinates, a' i e. an arbitrary continuous one-to-one corresystem
spondence of the set of numbeis (x and the points on the surface. By
means of a measuring-rod we can now measure the distance ds between
the points x and x l -\-dx Since then ds and dx are known numbers, the
?
equation (18) represents one equation for the determination of the
unknowns g lk Since the metric tensor in two dimensions has three
.
independent components we may thus by performing this procedure
for three suitable line elements starting from the same point x
completely determine the values of g lk at this point. This can now be
l
done for any point on the surface, thus obtaining a complete experimental determination of the metric tensor
In this way the geometry on the surface becomes an empirical science
subjected to the limitations arising from the limited measuring accuracy
Now imagine that we heat the surroundings of a gi von point on the surface
so that the measuring-rods are dilated
neglect this dilatation we shall, by
we
when inserted at this point. If
means of the method described
above, find wrong values for the components of the metric tensors.
Since, however, the thermal dilatation is different for measuring-rods
prepared from various materials, it is easy to correct for this error and to
find the 'real' values for the
components of the metric
tensor.
On
the
other hand, it is obvious that, if all measuring-rods in the neighbourhood
of a given point for one or another reason were dilated at the same rate
independently of the material of which they are made, it would be
impossible to observe this dilatation and in an unambiguous way to
no well-defined meaning in the statement that such an expansion of the standard rods has taken place, and
correct for
it.
Therefore there
is
VIII,
&
THE GENERAL THEORY OF RELATIVITY
87
233
from a physicist's point of view the metric tensor and the geometry
obtained by measurements with the natural standard rods must be the
'real' geometry on the surface.
All the considerations of this
paragraph for the two-dimensional case
now be immediately
can
The only
difference
is
generalized to spaces of 3, 4, or n dimensions.
that the points in an n-dimensional space are
by n coordinates x and that
now
can take on the values from
equations
l
characterized
element and the angle between two
by
(18)
and
n equations
equations
(19), respectively,
(30) or
line
all
1
indices in the preceding
The length of a line
to n.
elements are then again given
lines are defined by the
and the'geodesic
by means of the variational principle expressed By the
(21), (22), (29).
General accelerated systems of reference. The most general
admissible space -time transformations
88.
In
84
we have
seen that the spatial geometry in a uniformly rotating
is non -Euclidean and also the temporal description
system of reference
more complicated than in the systems of inertia. This may be regarded
an effect of the gravitational field present in the rotating system of
reference According to the principle of equivalence we must therefore
is
as
expect that a gravitational field
general will manifest itself not only
the
of
forces
presence
by
gravitational
(centrifugal forces, Coriolis forces,
gravitational attraction between masses, etc
of space and time measurements.
but also in the results
),
Let us start again from a system of inertia / with the usual space and
time coordinates (X, Y, Z, T) To each set of values of these variables
corresponds a certain event which is represented by a point in (3+1)space with coordinates
X =
1
These coordinates
that
differ
(X,Y,Z,cT).
we have dropped the
in the fourth coordinate.
sional line element (IV. 26) thus takes the
efo
= dX +dY*+dZ
2
where
Otk
(39)
from the coordinates defined
-c* dT*
- Glk dX>dX k
for
== k =- 1,2, 3
4.
Instead of the pseudo- Cartesian coordinates
for
l
general 'curvilinear' coordinates x in four-space
formations
/
l
,VL-\
k
x l (X
),
t
(40)
=k=
only in
The four-dimen-
form
for
in (IV. 2)
(41)
we shall now introduce
by means of the trans/A*\
(42)
THE FOUiNDATlOJNS Ob
234
where the xl
Xk
are arbitrary continuous
the variables (X
a special case of
The transformations
).
(42).
By
1 ), (3),
the relations reciprocal to (42)
dX*
Using
i-iri*
= ~~
dxk
88
10) obviously represent
we obtain
n^rl*
(d.^
we obtain
&
and differentiate functions of
differentiation of (42)
3~.i
U*'
From
Vlll,
Afc
dx k
in the
same way
(44)
(44) in the right-hand side of (43) gives
A}& k dxk
l
dx l
Since this equation
is
to hold for arbitrary dx l
we must have
=
*"^
for
In the same way, substituting from (43)
equations
A} A'A
Elimination of
dX
d8*
=-
in
(44),
we obtain the
(45')
Sj^
(40) thus gives the following expression for the
interval:
ffi*
(45,
k.
flu
0, m
g lk dx*<h*
^ Ar ^ A
(46)
A k~ ^ A
(47)
(Greek indices run from 1 to 3, Latin indices from 1 to 4).
The system of coordinates (JL determined by the transformations
I
determines a definite system of reference. Defining a 'point
of reference' as a point with constant values for the three space coordinates (x ), the system of reference corresponding to the system of
(42) also
coordinates (x ) can be defined as the collection of all points of reference.
In general such a system of reference will, of course, not be rigid, since the
1
different reference points may have largely varying velocities relative
to 7. The motion of the different points in the system of reference will
therefore in general be analogous to the motion of a fluid and we shall
confine ourselves to such transformations (42) for which the correspond-
ing system of reference can be pictured by a real fluid. This means that
the velocities of the points of reference relative to the system of inertia /
must always be smaller than
we
Since for any point of reference dx l
obtain from (44) for the velocity components v of such a point relative
to/>
c.
dX'
Aj
(48)
VIII,
THE GENERAL THEORY OF RELATIVITY
88
If this velocity
is
to be smaller than
V
= ~T
<
\
c
c
we must have
A4 A 4
or
c,
235
Aj Aj
<
0.
(49)
In order that the system of reference defined by
(42) may be physically
the
must satisfy the
admissible
transformations
realizable,
space-time
condition (49) which, on account of (47), involves
044
<
0-
(50)
At every point of reference we imagine a coordinate clock to be inserted
showing the time t
x*/c, and we must now further demand that the
time description obtained in this
way shall give a reasonable causal
of
description
physical phenomena. Thus a real signal which is emitted
from a point of reference (x l ) at the time t must arrive at a point of
reference (x^dx 1 ) at a time t+dt with positive dt. Since signals can at
most have the velocity
relative to /, the line element
must be smaller than or equal
time track of the
to zero for two adjacent points on the
In the system of coordinates x this means
l
signal.
ds 2
=g
lk
dx l dxk
0.
In other words, any two events which are simultaneous in the system
of coordinates (xl ), i.e. for which dx*
0, cannot be connected by a
so
that
in
must
have
this
case
we
signal,
k
g lk dx*dx
or, since
dx*
l
K
g lK dx dx
0,
This inequality must
>
>
0,
0.
(51)
now
hold for arbitrary dx which shows that the
dx^dx" must be positive definite. The necessary and
l
quadratic form g lK
sufficient condition for this to be true
is
that
all
subdeterminants of
the scheme of numbers g lK are positive.
The admissible transformations (42) must therefore be such that the
corresponding g lk satisfy the conditions
0u
<7
>0,
9"
9iK
t/Ki
VKK
in
U ).
013
0,
031
where
012
032
and K may be any of the numbers
<? 44
<
0,
(52)
033
1, 2,
3 (no
summation over
THE FOUNDATIONS OF
236
From
(52) it follows that the
determinant
012
013
014
022
023
024
031
032
033
034
041
042
043
044
011
=
If
I0,*l
88
VIII,
we introduce another system
<
0.
of coordinates x' 1
(53)
by means of the
transformation equations
x' 1
t'
dxl
x' l (x k )
aju
we have, by analogy with
dx k -=
k
&[ dx'
dx k
dx k
(54)
fa*
dx' k
(45), (45'),
Expressed in the new coordinates the interval
geneous quadratic form
= k dx' dx' k
- glm &[
ds 2 =- g lk dx dx k
l
where
are
new
g lk
again be a homo-
g\
(56)
cx
g'kl
functions of the coordinates
^=
(57)are
will
0; n
x'
1
.
The
(57)
relations reciprocal to
xr,
(58)
which can easily be verified by substituting from (57) in the rightside of (58) and using (55). The transformations (54) must be such
that the new functions g'lk again satisfy inequalities of the form (52).
hand
Tn general the system of reference R' defined by the system of coordinates (x/ ) will be different from the system of reference R corresponding to the coordinates (x ), but if the transformations (54) are of
1
the form
x
x'*
= x(x]
= x'*(x*) = /(a*)
where the space coordinates
x' 1 are
x* only, the systems of reference R'
(59)
functions of the spatial coordinates
R are identical. For in this case
and
the transformation simply implies another notation for the points of
reference in R together with an arbitrary continuous change in the rate
and
(x
setting of the coordinate clocks. While each system of coordinates
corresponds to one and only one system of reference R, we can
always in a given system of reference introduce an
infinite
number of
THE GENERAL THEORY OF RELATIVITY
88
VIII,
237
which are connected by transThe coefficients a*. and a k corresponding
different space-time coordinate systems
formations of the form
(59).
to the special transformations (59) obviously satisfy the conditions
(59')
In general the gravitational fields in different systems of coordinates
be different, but for physical reasons it is convenient to consider
will
the gravitational fields in all systems of coordinates connected by (59) as
identical, since all these systems of coordinates correspond to the same
system of reference. In different systems of reference, however, there
will in general be different gravitational fields. Jn this respect the various
systems of inertia are exceptional, since they
gravitational
89.
all
have a vanishing
field.
Space and time measurements in an arbitrary system of
reference. Experimental determination of the functions g ik
Let us now consider an arbitrary system of reference R into which we
have introduced a certain system of coordinates (x ). Consider in particular two points of reference A and B in this system with the space
coordinates (x ) and (x -}~dx ), respectively The spatial distance da
between A and B at the time /
JT*/C can now be measured by means of a
standard measuring-rod connecting the points A and B and at rest
In the limit of very small dx B will also be practically
relative to A
at it\st relative to the measuring-rod In order to express da in terms of
the functions g lk we introduce the system of inertia 7 relative to which
l
----
the point of reference A (arid approximately also B) is at rest at the
time t If
denote the pseudo-Cartesian space-time coordinates in 7,
the transformation from 7 to R is given by the equations (42)-(48).
However, since 7 is the rest system of the point A at the time considered,
we have, on account of (48),
A^ (60)
at the point
and the time
t.
The
differences
dX
between simultaneous
values of the Cartesian coordinates of the points A and
the measurement are obtained from (44) by putting
i
~-
Hence
and
dX
dx*
dt
= A K dx.
B at the time of
0.
(61)
A 4 dx l in (44) would vanish anyhow, even if
which
means
that
the values of dX would be approximately
/ 0,
the same as for positions of A and B which are simultaneous relative to 7.
On
dx*
account of (60) the term
THE FOUNDATIONS OF
238
89
VIII,
According to the general assumption formulated on p. 223, the
standard measuring-rod in the system R has the same length as the
2
measuring-rod in /; therefore, da is simply
Using the expression
dX we
in the differentials (dx L ),
da 2
see that
(61) for
is
a quadratic form
i.e.
= y dx'dx"
y = A* A*.
da 2
with
(62)
lK
lK
Now we
get from (47)
and
^-
4
A
A,_
Hence
(60)
where we have put
EE
g '4
~~J^
Furthermore, we obtain from (47) for
\A
t,
g lK ~\-A l
(63)
&
AK
which leads to the following expressions for the spatial metric tensor
YlK
QlK
+ YlYK:
The metric tensor y lK which determines the
spatial
64 )
geometry in the
will thus in general not simply be equal to the spatial
reference system
part g LK of the four-dimensional metric tensor g lk This is the case only if
.
0*=0
or
0.
(65)
In a system of coordinates where the equations (65) are satisfied at
every point in 4-space, the time axis is everywhere orthogonal to the
spatial coordinate curves. Such a system will therefore be called timeorthogonal.
Now
C inserted
consider a standard clock
point A of the system of reference R.
of this clock is then given by
ds*
since
by
is
dx
for
a clock at
rest.
44
at rest at a given reference
The line element of the time track
(dx*)
While
(66)
x*/c
denotes the time shown
the coordinate clock at A, the increase dr in the time of the clock
given by
^=_
c,
VIII,
THE GENERAL THEORY OF RELATIVITY
89
This follows at once from the in variance of ds 2
37).
(cf.
if
239
we introduce
the system of inertia 7 which is momentarily at rest relative to the clock
C. Expressed in terms of the space-time coordinates (X Qi ) of this system,
^ _ _^
becomes
ds*
since
dX Ql =
(d?70)2
Further, since the time
0.
7, we have
clocks at rest in
dT
drl
dr
-<7 44
TQ
given by standard
can be written
itself is
and
(66)
dt*.
(66')
Consequently the function ^/(~g^) determines the ratio between the
C and the coordinate clock at the place con44 may thus be obtained experimentally by
rates of the standard clock
The quantity </
measuring the ratio of the rates of the two clocks in question. If we do
this at each reference point and at all times, we get gu (x l as a function
sidered.
of the space-time coordinates (x l ).
The metric tensor y iK which, by (62), determines the space geometry
in S can now also be obtained by direct measurements, applying a
method
similar to that explained for the two-dimensional pase in
To determine the
87.
we simply need
six
independent components y lK
measure the lengths of six properly chosen hne elements (dx
to
at each
point and at every time.
Now, if we could also find a procedure which would allow us to measure
the quantity y the metric tensor g lk would be completely determined To
.
t ,
purpose we consider a light signal which starts at the reference point
with space coordinates (x l ) at the time t and arrives at a neighbouring
point B: (x -\-dx ) at a time t-{-dt, say. The time track of this signal is
this
characterized
by the equation
ds*
for in the
=g
lk
dx l dx k
0,
(67)
system of inertia / the velocity of the signal
is c,
which
means that the invariant
ds 2
The equation
=O
lk
(67)
may
g iK
or,
by means of
dXidX*
dX*+dY*+dZ*-c* dT*
If
we
divide this
0.
also be written
dx'dx+2g t dx^dx^+g^
L
(63), (64),
and
(dx*)*
dx-)+^(-g^} 7l dx^dx^+g^
2
equation by dt we obtain
w*
0,
(62),
da*-(<y dx^(y K
L
{yt
^_cV(-<744)}
(dx*)*
0.
2
,
(68)
THE FOUNDATIONS OF
240
where the
dx
-
wn
89
VIII,
(69)
at
m the direction
are the components of the velocity of the light signal
n and
l
w=
= V(y^'^)
Tt
(69')
the magnitude of this velocity. In the first place, we see from (68) that
w depends on the direction of propagation n of the
is
the light velocity
if
signal
In
fact,
in the
system of coordinates considered.
we
using (69) in (68)
where
ylK nlnK
On the other hand,
if
get
*>
1
-
(yt^
we solve the equation
<
(70)
(y L n
with respect to
),
a measurement of the velocity of light in three different directions n
In this way it is
will allow us to determine the three quantities y
determine
possible in principle to
the complete metric tensor g lk
The
90.
spatial
geometry
>
all
the quantities y llc
</ 44
i.e.
by experiments.
in the rotating
system of reference
Considering again the rotating system of coordinates introduced in
we obtain from (1), (2), (3), and (10) the transformation equations
(42), or rather the corresponding reciprocal equations, in the form
84,
X=
rcos(#+o>Z),
Z^z,
By
differentiation
Y=
T=
rsm(&+cot)
t
(71)
and introduction of (dX,dY,dZ dT) into the exy
pression for the interval
we
get
ds 2
= dr*+r
Thus we have
^ +d2 +2cur
2
d&dt-(c*-rW)
dt*
=g
lk
dxl dx k
(72)
VIII,
THE GENERAL THEORY OF RELATIVITY
90
241
other components of g lk being zero. Hence, from (63) and
all
yt
=
=
y lK
OJ
for
~
2
r2
|0,
(64),
8 l2
(74)
-^ K
Thus if we calculate the spatial line-element da 2 by means of (62) and
(74) we come back to the expression (7). Further, (66') together with the
expression (73) for
j/ 44
leads back to the equation
(9).
The space geometry defined by the line-element da 2
non-Euclidean. In the plane z = Z
have only the two coordinates (x l ,x 2 )
K
y lK dx dx
l
is
on the rotating disk, we
(r,$) and the geodesies are
0, i.e.
determined by the equations (32)
da l
da da
2 dx l
dx dx K
l
They
define the curves of shortest distance as
rods at rest on the rotating disk.
In our case
measured by measuringwe get from the second
equation (75 a) and (74)
d
and by integration
1
where a
is
a constant, and &
r
rz
ST^#
w 2 /c z
(76 a)
>
Hence,
d&jdv.
- --
fl-rW/c*\
Oil
(7e
,,
'
1'
'
Further, from (756)
r2
dr
TT
Hence,
TO
r
-*
which by integration gives
3595.60
"
i-^^-S
r as a
*
\
_
T-?V
r2
->
function of & for the geodesies.
/77
7T
(
THE FOUNDATIONS OF
242
If the constant of integration a
radius vectors with
is
zero,
90
VIII,
we get from (76 a') &
= 0,
In Fig. 17
constant are geodesies.
i.e.
the
we have
given a picture in a Euclidean plane of some of the geodesies on the
rotating disk. In this picture the points on the disk with the coordinates
(r,
&) are depicted as points with the polar coordinates
(r,
#).
The curves
FIG 17
in this picture are therefore simply the curves in the fixed JfF-plane
with which the points on the geodesies of the rotating disk coincide at
T
0. The radius r* of the disk is defined by (4), i.e.
the time t
r*
For the geodesies
starts at the point
sponds to an a
(78)
c/co.
OB and OA we have a =
(/*
0) at right angles to
0.
OA,
The geodesic AB which
i.e.
with
0, corre-
which follows from (76 a'), (766):
2
r
/
_
(
-r
_
~
__
r
__
a>'c)
With increasing values of & the radius vector r
increases also with a rate
determined by (77), and it is seen at once that dr/d&
limit c -> oo, where we should have
dr
-
The broken straight line in
larger than in the
a 2 )*
r(r
-
air
is
Fig, 17 is the curve
(80)
determined by (80) which
the geometry on the rotating disk were Euclifrom (76 a'),
dean. When r approaches the value r*
c/a> we get # ->
as tangent at the point B.
which means that the geodesic A has
would be the geodesic
if
Consider two geodesies x\
a:i(or 1 ),
OB
xfa
^(o^) which go through
THE GENERAL THEORY OF RELATIVITY
90
VIII,
243
between the two curves at the point of
the same point. The angle
intersection is determined by (19). Thus, on account of (74) and (76),
a
cos 6
= >W di
^-=
Ct)
1C
n~'" Z
-=*.}
'-
r2
(81)
where
c^
and
are the values of the constants a for the
two geodesies
This expression can be used everywhere except at the
0.
point
Now consider the triangle OAB. At O the angle between the two
of the
&B
sides of the triangle is equal to 6 O
\TT, for in the centre
in question.
disk the geometry is Euclidean.
A B at A
is
<
The angle between the geodesies AO and
found from (81) by putting
thus,
A = \7T.
Finally, we get the angle 6 B between the geodesies
i.e.
(81)
by putting
0,
a2
Hence the sum of the angles
i.e.
TT.
-a, r
r*
in the triangle
c/a>.
BA and BO from
Thus we get
OAB is
sum of the angles in a triangle on the rotating disk is smaller than
Only if the triangle is close to the centre is the sum of the angles
the
approximately equal to the Euclidean value TT. There are even triangles
on the rotating disk for which the sum of the angles is zero. If, for in-
#B = |TT, we see that the triangle CBD formed by the geodesies
CB, BD, and DC has a vanishing sum, i.e.
stance,
QCBD
Thus the sum of the angles in a triangle on the rotating disk can have all
values between
and ir. The spatial geometry as determined by observers on the rotating disk is the same as on a surface of negative
curvature in a three-dimensional Euclidean space.
THE FOUNDATIONS OF
244
90
VIII,
Sometimes
it is convenient to use a different set of space coordinates
the rotating system connected with the cylindrical coordinates (r,#, z) by the equations
(x,y,z) in
rcos#,
r2
rsm&,
x 2 -{-y 2
(82)
In these coordinates the line element (72) takes the form
ds 2
dx 2 +dy 2 +dz 2 +2aj(-ydx + xdy)dt-
The time tracks
91.
of free particles
and
/l-^c ^
2
(83)
light rays
Consider a material particle which is moving freely under the influence
solely of the gravitational fields in an accelerated system with the
coordinates (x l ). Since these fields are supposed to be non-permanent,
they can be transformed away simply by introducing the pseudo-
of the system of inertia / from which we
88. In this system the motion of a free particle is uniform,
Cartesian coordinates
started in
i.e. its
time track
is
(X
a straight line defined by the equation
where A
is
an arbitrary parameter. In other words, the time track of a
freely falling particle is a geodesic in 4-space. The geodesies are defined by the variational principle (21), (29), (33), where the indices &, k
now
are running from 1 to 4. If we introduce the proper time r
s/ic
as parameter, the vanational principle (33) states that the variation
f
--
dr
must be
zero for
all
dr
(85)>
(
2 dx* dr dr
variations 8x l which vanish for r
r1
and T
r2
This leads to the Euler equations (32) or
d( 9lk dx
dr\
dx*dtf
\_ldg
2^-fo dr*
~&r)
dtfdx*
kl
9lk
^^ _
~~
'
b)
In a pseudo- Cartesian system of coordinates the metric tensor is
as defined by (41). The equations (86) then reduce to (84), and (85)
Glk
reduces to the equation (V. 102) used in 59.
Now consider a light ray in empty space. In the system of inertia /
its time track is again given by (84), but with the extra condition that
ds 2
= Glk dX dX k =
l
0.
The time track of a light ray is thus a geodesic of
THE GENERAL THEORY OF RELATIVITY
91
VIII,
245
and we can therefore not use the length as a parameter.
we thus have in an arbitrary system of coordinates (xl the
equations (30) and (31) with the constant in (31) equal to zero, i.e.
zero length,
Instead of (86)
dx k \
dt
__
dtif* d\)
k
l
dg kl dx dx
2W
where
rfA
==
xl
dA
a;
~
__
'
~d\~dX
(A)
be any parametric representation of the time track.
may
92.
dxl dx*
9lk
'
The dynamical
gravitational potentials
we have seen that the gravitational field in an arbitrary
89 and 90
In
accelerated system of coordinates (x l ) influences the space and time
measurements as made by standard measuring-rods and standard clocks.
The space geometry, for example, is determined by the spatial tensor y tK
defined by (64). We shall now determine the quantities which describe the
dynamical action of the gravitational fields. For this purpose we shall
use a test body of arbitrary mass, which is placed at rest at the point
of our system of reference at which we wish to measure the gravitational
field. The acceleration imparted to this particle by the gravitational
then determines the strength of the
field
From
'
= =
*
(86)
we
get, for a particle
1. 2, 3.
d *x*
which
field.
is
dx*\
momentarily at
rest,
taking
dgjdx'
(88)
Further,
we get by means of (64),
motion
c2
where
is
dr 2
ds 2
(63),
and
(62) for a particle in arbitrary
L
K
g lH dx dx +2g^ dx
dx*+g^
(dx*)
the proper time, and the
are the components of the velocity of the particle.
Dividing (89) by
gives
dx*
~dr
(dx*)
-c
and solving this equation with respect to dx*/dr
/
(-M
v u'
:^Vl)
<v(
044);
- U2/C2 }-*
\
J
'
(90)
THE FOUNDATIONS OF
246
is
92
da
where
VIII,
~Ji
at
the magnitude of the particle velocity.
Hence
and
g^
for a particle
= cy
which
is
,~
momentarily at
^ 2 /c 2{744
-f-
rest
we
>
get after
some
cal-
and
Using (91) and
(90) in (88)
we thus obtain
inn\
(92)
i
-
The
left-hand side represents the gravitational acceleration imparted to
the test particle,
being the covariant components of the acceleration expressed in the
curvilinear coordinates of our system of reference (see 100). Thus the
dynamical action of the gravitational
044 and y
t
If
we
described by the functions
field is
we put
get from (92)
44
= ~^dx
/i
+
!
AJ \
+ -c
(94)
}^-
(95)
/ dt
On account of the analogy of this expression with the expression for
the electric force on a charged particle at rest in terms of the electromagnetic potentials, the quantities x an d y L will be called the gravitaand vector potential, respectively. The scalar potential
1 of the
X has been normalized so as to give 44 the value
special theory
tional scalar
<jr
of relativity for a vanishing potential.
If y l is time-independent, the gravitational acceleration
equal to the gradient of the scalar potential
is
simply
at
=-
3
.
(96)
This
THE GENERAL THEORY OF RELATIVITY
92
VIII,
is,
for instance, the case in the rotating
sidered in
90,
and from
and
(73)
= -r
we
(94)
co
247
system of coordinates con-
get in this case
(97)
Thus the gravitational acceleration of a particle of zero velocity lies in
the direction of increasing r and is equal to ro> 2 This is in accordance
.
with the usual expression for the centrifugal
force.
The rate of a moving standard clock in a gravitational field
From (90) we get for the proper time of a particle moving in the gravi-
93.
tational field described
by the
potentials (y t x)
,
Since r
is
the time measured by a standard clock following the particle
moving standard clock compared
in its motion, (98) gives the rate of the
with the rate
dt of the coordinate clocks of the
the system of coordinates
is
time-orthogonal,
system considered. If
= and
we have y
(99)
The formulae (98) and (99) are the generalizations of the formula (II. 38)
and express the retardation (or advancement) of moving clocks in the
case where gravitational fields are present.
For a clock at rest in our system of reference
we have
(100)
with (66') and (94). The rate of a standard clock thus
depends on the scalar gravitational potential at the place where the
in accordance
clock
On
is
situated;
it is
lower at places of small gravitational potential,
we have, according to (100) and (97),
the rotating disk
dr
+ 2x/c a
d^/(l
(101)
dtj(l-r*a>*/c*),
a standard clock far from the Centre has a slower rate than a standard
T. This
clock placed at the centre which simply shows the time t
i.e.
>
retardation of a standard clock at a place with r
will be differently
7
in
the
fixed
observers
and
the
in
system
rotating system
interpreted by
An
observer in / will explain the retardation by the motion of the
but
particle. In this system we have no gravitational field, i.e. x
is
u
of
/
ra>. An application
thus leads
the velocity of the clock
(99) in
S.
to the formula
drQ
dt<J(
u*/c*)
-=
dt<J(
rW/c
2
).
02)
THE FOUNDATIONS OF
248
On
the other hand, in
field
(102).
^r
An
o>
2
,
and
observer in
of the gravitational
94.
the velocity u
93
but we have a gravitational
same expression (101) or
thus explain the retardation by the action
present in the rotating system.
(99) again leads to the
will
field
Transformation
0,
VIII,
of coordinates inside a fixed
system
of
reference
Let
be a set of space-time coordinates corresponding to a certain
system of reference jR. By the transformation (59) we may then introduce new space-time coordinates inside the same system of reference R.
(x
(59) simply introduces a new numbering of the reference points in R together with an arbitrary change in rate and setting
of the coordinate clocks This, of course, cannot give rise to any change
The transformation
geometry in R determined by measurements with standard
measuring-rods, i.e. da as defined by (62), (64), and (63) must be invariant
in the spatial
by the transformation
Appendix 4.
A formal proof of this statement is given
(59).
in
Since the system of reference is unchanged by the transformation
(59), the gravitational field must also be regarded as unchanged. The
gravitational potentials (x,y
defined
be transformed in accordance with
of that kind
(57).
and
(63) will, however,
In this respect a transformation
by
(94)
thus analogous to the gauge transformation (V. 23) of the
electromagnetic potentials by which these potentials are changed without
is
any influence on the electromagnetic field derived from the potentials.
In many cases it is possible by such a 'gauge transformation' of the
gravitational potentials to give the potentials a particularly simple form.
first place, it is always possible by a transformation of the type
In the
x''
1
;
z' 4
x'*(x*)
= f(x*)
(103)
We
need
to ensure that the scalar potential in the system (x' 1 ) vanishes.
the
new
coordinate
so
that
time
variable
t'
the
new
choose
x'*/c
only
clocks have the
and
same rate
(100), this is
as standard clocks at rest.
On account of (66')
obtained by putting
t
V(-
(104)
where the integration is performed for constant values of the space
coordinates (X ), and ^(x ) may be any function of the space coordinates.
With this choice of the new time variable we get, since (100) must hold
L
THE GENERAL THEORY OF RELATIVITY
94
VIII,
also in the
249
new system,
dt'
=--
dr
dt'
III
V
i.e.
x'
=
+ -*-)
c
dt'J(-<fu ),
fik=-l.
0,
(105)
A more useful simplification would, however, be obtained if the vector
potential could be 'transformed away' by a transformation of the type
considered, for this would mean that the new system of coordinates
is
time-orthogonal and
all
formulae are considerably reduced
Let us
therefore try to find a transformation (103) such that
y\
0.
(106)
Since the interval can be written in the form (89) in every system of
2
coordinates, and since da is invariant under the transformations (103),
we must have
^.'4
must be the
!/p^ _
4
rt
(107)
total differential of the function /(x' ) in (103).
Thus we
see
that the space-time derivatives of the function /must have the following
ratios
j-f
This
is
J^ J
?i_
__
-2^?
73 __
__!
QAON
'
equivalent to the validity of the three equations
for
i=l,2,3.
V
(109)
;
These conditions could also be obtained from (106) by mep,ns of the transformation properties of the gravitational potentials (see Appendix 4).
Now the simultaneous differential equations (109) have a solution
only when the following compatibility conditions are
multiply (109) by the operator
and subtract the equation obtained
in this
satisfied.
If
we
way from the equation with
THE FOUNDATIONS OF
250
and K interchanged,
the quantity
we
VIII,
94
get after a simple calculation the condition that
1-
must be
zero,
o> tAC
i.e.
(HI)
for all values of t and K. (Ill) together with (110) represent the general
condition which the dynamical gravitational potentials must satisfy in
order that the vector potential may be made to vanish by a transforma-
tion of the type (103).f
In the case of the rotating system of coordinates considered in 90,
we see at once from (110), (73), and (74) that the only non-vanishing
components of
o> tK
are
aj l2
o> 21
r^_-_,
(112)
>
0. The condition 1 1 1 is therefore
it is not possible by a simple
and
system
which are different from zero for
not fulfilled in the rotating
change of rate of the coordinate clocks to introduce a time-orthogonal
system of coordinates in this system of reference.
The gravitational field in a given system of reference R is called
stationary if it is possible, by a suitable choice of the space-time coordinates in R, to ensure that
all
components of g lk
are independent of the time variable.
i.e.
If simultaneously
y LKJ
#,
and y
we can obtain
0, the gravitational field is called static. The gravitational field
t
in the uniformly rotating system of reference is thus stationary.
95.
Further simple examples of accelerated systems of reference
Let
X =
1
(X,Y, Z,cT) again be pseudo- Cartesian space-time coordinates in a system of inertia /. The Galilean transformation (I. 2)
y
b
-*\.
7
wT
V
J.
it
)
if
JL
A>
7i
J,
f
I
JL
n 1 *U
\LlOf
then defines a new system of reference which obviously is the system of
inertia /' moving in the direction of the
-axis with the velocity v relative
to /; for each reference point (x>y, z)
constant is moving with the
X
=
f This condition was found by Weyssenhoff, see J. v. Weyssenhoff, Bull. Acad.
Polonaise. Sor. A, p. 252 (1937) soo also Z8. J. Phys. 95, 391 (1935) ibid. 107, 64 (1937).
;
THE GENERAL THEORY OF RELATIVITY
95
VIII,
same
ds*
i.e.
all
If
v.
velocity
02 2
we put
(x
k
g lk dx*dx
g 83
251
= Oik d
U
1,
==
41
v/c,
44
-(l-i>
other components vanishing. The system of coordinates
fore not time-orthogonal
form
(x,y,z,ct) the interval takes the
and from
and
(63), (64),
(94)
we
for
V33
/c
(a;*) is
),
there-
get
Since the gravitational potentials are constant the gravitational field
/', and the potentials may be
determined by (95) is, of course, zero in
transformed away by a transformation
Introducing
new
coordinates X' 1
X'
(59)
with / of the form (104).
f
(X \
Y',
Y'
'-
Z',cT
by
*Z'
11
(116)
r
we
A-
et
The coordinates (X /x are pseudo- Cartesian coordinates in 7 they
connected with the (X 1 by the special Lorentz transformation.
As a further example we consider the accelerated system
)
are
(x
defined
(x, y, z, ct)
by the transformation
X=
x+\gt*,
y,
T=
z,
t.
(118)
Each reference point in the system (x ) has a constant acceleration g in
the direction of the JT-axis relative to /. A simple calculation gives
1
ds*
dx 2 +dy*+dz*+2gt dxdt-c*
dt*(l
-^1
i.e.
all
ffu
= fe = 033 =
!>
(7l4
= 041 = ^/C,
other components g tk being zero.
044
c2 /
1
f
""
(119)
THE FOUNDATIONS OF
252
VIII,
95
Hence,
Yi
fl*_
-.0,0),
=-
V
(120)
733
^ r
This system of coordinates corresponds to a physically realizable system
T c/g, for only in this case will the velocity
of reference only for t
0.
the
reference
of
gt
points relative to / be smaller than c and gr 44
<
<
The space geometry in this system is defined by the spatial line element
da 2
It
y lK
Jr 2
dx<<dx
-J-
+ dif+dz
(121)
g*t*/c*
non-Euclidean on account of the Lorentz contraction of the measur-
is
ing-rods in the moving system Since the potentials do not depend on the
space coordinates the quantities w lK in (110) are zero, which means that
the vector potentials can be made equal to zero by a transformation
(103). Integrating the equations (109) we find that the transformation
leads to the desired result,
the
i.e. if
ds 2
is
written in the form (89)
we
get
</44
=-
12^2
J
'
=~
e 2gXIc*
in^yaZTa
y
'
(123)
^9/
/
It
new coordinates
is
easily verified
'
by
'
1 \
'
direct Calculation that (122')
is
identical with
(119) when use is made of the transformation equations (122).
The gravitational field in the accelerated system considered
static.
According to (121) even the space
is
non-
is
geometry
time-dependent,
the distance between two neighbouring reference points depends
on the variable t. This is also evident, since the measuring-rods in the
i.e.
accelerated system on account of the increasing velocity gt relative to /
undergo an increasing Lorentz contraction. Although our system of
will
reference
is
rigid
from the point of view of an observer in
/,
an observer in
the accelerated system itself will find that the system of reference points
is dilated in the direction of the #-axis,
THE GENERAL THEORY OF RELATIVITY
95
VIII,
253
For small values of t and x, i.e. if we retain only terms of the first order
in gt/c and gx/c z we get t - t(l-gx/c*), g^ = -(l+2gx'/c*) x = 9*',
and from (96)
o
a
(124)
'
'=-0i=<-0>')'
the gravitational field
i.e.
96. Rigid
constant in this region.
is
systems of reference with an arbitrary motion of the
origin
A system of reference is called rigid if the distance
between two
refer-
ence points, as measured by standard measuring-rods at rest in the
system, is constant in time. Thus the uniformly rotating system discussed in 90 is rigid, while the system considered at the end of 95 is not.
Consider
now a particle in arbitrary motion relative to the system / with
the coordinates X^
by the equations
(X,Y>Z,icT).
Its
time track
may
^ =/,(*),
be described
(1^5)
T being the proper time of the particle. We shall now try to introduce
a system of coordinates (x )
(x, y, 2, rt) which is the relativistic analogue
l
of a classical rigid frame of Cartesian axes following the particle in its
motion, so that the particle is constantly situated at the origin of this
frame of reference, and the space axes have constant directions. In
46
we have determined the successive systems of inertia
momentary
systems of the particle
rest
$'(r) which are
and which are successively
obtained by infinitesimal Lorentz transformations without rotation of
the spatial axes. The transformation from the fixed system (XJ to the
coordinates x[ in S'(r) is then given by (IV. 140), where the coefficients
a lfc (r) are determined by the differential equations (IV. 136, 137).
Now
defining the system (x
x* =- x't
in (IV. 140),
in the
we
system
by putting
0,
(126)
find that simultaneous positions of the reference points
l
(x
at the time
coincide with the simultaneous positions
of the reference points in S'(r) at the time x^
0, and the coinciding
reference points have the same values for the spatial coordinates in these
two systems of coordinates. The transformation connecting the variables
l
(X and (x ) are thus
%)
X
where the
=/&)+**(*)>
coefficients a lk are the solutions of (IV. 136).
are completely determined
by the given motion
127 )
These equations
(125) of the particle
THE FOUNDATIONS OF
254
which
lies
permanently at the origin x
by means of
tiation of (127) gives,
dXi
we
If
VIII,
96
of the system (x 1 ). Differen-
(IV. 136),
= [AW+a^W] dt+dx*Kl
(t)\
use (128) in the expression for the interval
(128)
we
get,
on account
of (IV. 138),
ds 2
where
= dX dX = dx*-\-dy
= (x^ct) = (x,y z,ct)
=* U=
gK = gK
l
(t)
Kl
* Kl f
30)
are functions of t only, which are completely determined by the motion
of the origin of our system of coordinates (x 1 ) relative to the system (XJ.
The quantities g K are equal to the components of the acceleration of the
momentary rest system S'(t) (cf. (IV. 42, 42')).
The system of coordinates (x defined by (127) is time-orthogonal.
The corresponding system of reference is rigid, for the distance between
two reference points (x,y,z) and (x-\-dx y-\-dy,z-{-dz) is given by
particle in its
da 2
dx*+dy*+dz
(131)
Thus the space geometry is even Euclidean and (x, y, z) are Cartesian
space coordinates. The vector potential is zero and for the scalar
potential we get
(132)
dependence on the space variables is thus always of the same type,
only the coefficients g = g(<) will be different for different motions of
the origin. The gravitational field is determined by (96) and (132):
Its
Hence,
in
2
--gradx - _g(i + (g. x )/c ).
a region around the origin, where
(133)
(a.xKc,
the gravitational field
g(<)
homogeneous, the gravitational acceleration
a
being function of t which is uniquely determined by
is
the motion of the origin x ?=
relative to /.
The rate of a standard clock at rest at the point
(132)IS
At the origin x
is
dr0== ^(l
a standard clock.
we have TO =
t,
+ (g.x)/C
i.e.
2
).
x as given by (100) and
(134)
the coordinate clock at this place
It
THE GENERAL THEORY OF RELATIVITY
96
VIII,
may
255
be shown that the type of accelerated systems considered in
with the rotating system of 90 are essentially the
this section together
only possible rigid systems of reference in the case of non-permanent
gravitational fields.
97, Rigid
frames
of reference
moving
in the direction of the
X-axis
the origin O of the system (x ) is moving in the direction of the
Jf-axis of the system (XJ, the coefficients a lk are given by (IV. 154, 147),
and the transformation equations (127) reduce to the equations
When
X=
smh 6 dt+x cosh 0(t)
(135)
smhB(t)
J
o
which are
also obtained
the vector (130)
we
get,
from (IV. 155) by the substitution
by means of (IV. 154, 146),
g-
For
(126).
d
(0,0,0),
g(t)^c /->
(136)
at
Hence, from (129),
(137)
This
is
also easily obtained directly
by
differentiation of (135)
and
substitution in the expression for ds 2 in terms of the differentials
(dX^, In this case the gravitational field is parallel to the x-axis. The
conditions (52) are satisfied everywhere except on the plane x
where gr44 becomes equal to zero.
In particular,
if
the motion of the origin
the constant rest acceleration
i.e.
g(t)
at
In this case the gravitational
being
g,
is
c 2 /<7,
a hyperbolic motion with
we have, according
=g=
to (IV. 159),
constant
field is static,
the gravitational potential
THE FOUNDATIONS OF
256
The transformation equations then reduce
X=
97
to
- l\ + x cosh g-
- (cosh ge
g\
Y =
VIII,
(140)
y,
9
as seen
By
from (135) and (138) or from (IV. 160) and (126)
elimination of the variable
in (140)
we
get
X = {[(l+^)+^W-J}|
?
Z
Y~y,
Thus we
for x,
?/,
see that, relative to /, each reference point with constant values
z performs a hyperbolic motion in the direction of the ^-axis
-- x, Y
z at T
?/, Z
starting at the point
The acceleration in tlus hyperbolic motion is
29),
and the velocity
v -~
on account of
dX
dT
at the
time
[(!-)
yx/c
n
|n
(143)
,gt
tanh
.
2
\
is
2 2
)
(142)
yx/c
gT
-
with zero velocity
f/
1 -f
(see
0,
(/
-,
T 2 /c 2 Y-
c.
(140).
The velocity of the reference points relative to 1 thus depends on x,
and from the point of view of an observer in / the system of reference R
corresponding to the coordinates (x will not appear as rigid The distance between two reference points (x, //, z) and (x-dx, y, z) measured by
l
an observer
in
is
found from (141) and
j
__
(143).
7
__
"
//,
V(
-,2/ r 2\ ,/ r
'
C08h"to//C)^
(144)
the velocity relative to / of the system of reference R
v(T)
at the place considered. From the point of view of an observer in /, each
where v
is
part of the system
formula.
R is thus contracting in accordance with the Lorentz
For small values of / and x, where w e can neglect terms m gx/c 2 and
gt/c of higher than first order, the system of reference R considered here
is identical with the
system considered at the end of 95.
r
THE GENERAL THEORY OF RELATIVITY
97
VIII,
257
now
consider the motion of a free particle in the gravitational
field of the system (x* ). The time track of such a particle is determined by
Let us
with glk given by (137), and for
(86)
d x
/,
CiT
If the particle starts
last
1, 2,
3 (86)
becomes
gx\
C
(dt\
\dT/
from a point on the #-axis with zero velocity, the
equations (145) give
_~
__
From
(99) and (139) we get the following connexion between the
time
of the particle and the time variable t of the system (x l ):
proper
dr
-u 2 lc
2 2
dt^[(l+gx/c
where
(146)
],
dx
dt
the velocity of the particle,
is
(146) is equivalent to the last equation
(86).
Using
means of
instead of r as independent variable in (145),
2g/c
dt 2
[dx\
l-{-gxlc \dt)
*
easily verified
The
solution of this equation corresponding to the initial conditions
^
is
by
r
X
as
get
(146)
d 2x
The
we
XQ,
dx
-=
\J
-,
1U1
__
M48\
1* /
V
\J
= ?f( + ? ^-ir-l'L
1
by
differentiation
and substitution
velocity of the particle at the time
dx
<
149 )
in (147).
is
di + 03D
I
dt
Thus with increasing
&
dt
c*
the velocity increases, reaches a
amax
/c
_ c(l+gx
S
maximum
2
)
'
and decreases again to zero for t-+oo. If x > c 2 /g, the velocity of the
particle assumes values which are larger than c; however, u is always
smaller than the velocity of light, which is
w=
on account of
3596.60
(70)
and
c(l+gx/c
(137).
2
)
THE FOUNDATIONS OF
258
For
system
VIII,
97
-> oo the particle approaches the singular wall x
c 2 /g of our
of coordinates. At this place also the velocity of light tends to
zero,
and no
time
any kind will ever reach the boundary plane.
Using the expressions (149) and (150) for x and u in (146) we get
by integration for the proper time r of the particle at the coordinate
signals of
^o)
__<?!_
c*) ) coshV/c
/5
+ ?o]tanh?-'.
c
\g^
(151)'
V
c)
1
Finally, if the origin of the system (x ) is moving with constant
0, and the vector
velocity v in the direction of the positive X-axis,/,
</ t
defined
by
be, since the
30)
is
zero.
Thus the gravitational field is zero as it should
system of reference
moving with the velocity
the transformation (127)
formation in this case.
a system of inertia
Ul r,
constant and/
in this case
is
Further, since Ul is
identical with the special Lorentz trans-
v.
is
The clock paradox
We are now in a position
98.
to state the complete solution of the clock
paradox which was mentioned in 20 and which played a certain part
in the early discussions on the consistency of the theory of relativity. f
Consider two standard clocks Cl and (72 originally situated at rest at the
origin Oj of a system of inertia
At the time
*S\
with the space-time coordinates
the clock
is accelerated by a
(X, 7, Z, T) (Fig 18).
constant force F in the direction of the positive Jf -axis. When C2 has
reached the point A it has attained a certain velocity v, and from this
point on
C2 is allowed to continue m a uniform
velocity v until
it
reaches the point B, where
it
motion with the constant
meets a constant counter-
same magnitude F as before, but with opposite direction. C2 is
brought to rest at C and accelerated back to J5, having then attained
the velocity
v. Between B and A it moves with the constant velocity
is attacked again by the constant force F which
and
at
A
it
v,
brings
it to rest at O x
Let A'7 A"2 A'" T be the times which C2 takes to travel
the distances 0/1, AB, and BC, respectively. For symmetry reasons
the motion on the way back from G to A must be just the reverse of the
motion from A to (7, and further we must have
force of the
&"T -
A'7
Ann d. Phya. 17, 891 (1905); P. Langevm, Scientia, 10, 31 (1911);
Laue, Phys. ZS. 13, 118 (1912), H. A Lorontz, Das Relativitatepnnzip, 3 Haarlemer Vorlosungen, pp. 31 and 47, Leipzig, 1914, A. Einstein, Naturiu 6, 697 (1918),
C. Mollor, Dan. Mat. Fys. Mldd. 20, No. 19 (1943)
t A. Einstein,
M.
v.
98
VIII,
THE GENERAL THEORY OF RELATIVITY
259
Let 4r x and Ar 2 denote the measurements on the clocks C^ and C2 of the
time elapsed between the two encounters of the clocks, T X and r 2 being
the proper times of C and <72 respective^. Since C is constantly at rest
at the point 0, Ar t is equal to the total increase AT in the time variable
T of the system S1 between the two encounters. Thus we have
,
A Tl
- AT -
2(A'7
+A"T+A'"T)
2(2A"F+A"T).
(152)
Via
Similarly
18
we get
denote the increase in proper time of C% during its travel
through OA, AB, and BC, respectively The motion of the clock C2
from O to A is a hyperbolic motion and is thus described by the equation
where
r'2 , rJJ, T'%
(III.47),ic.
*=!
ll
f7lL
/.J1V211
\c
^
*-l,
/ J
(164)
where
and
r/?
is
the rest mass of 6V
dX
Hence we get
or
The
velocity
gT
dX/dT
is
thus
(155)
(156)
(157)
THE FOUNDATIONS OF
260
VIII,
Using (155) we can now calculate the proper time
the formula
of inertia
(II.
r'2
of Cz
98
by means of
38) of the special theory of relativity valid in the system
Sv Hence
AT
On account
AT
of (157) this
also be written
may
q&T
vie
Slnh
158 )
--
=-
tanh
or
/lr o\
Sinh
(1 58')
./(nIn the same
way we
get from
(II.
38)
v 2 / c2 )-
Now,
if
we apply a
for constant value of v
larger
and
159 )
larger force
the acceleration g
Fjm will also increase. In the limit g -> oo for
tt"T and r'2
constant v we see from (158) that both
r,J tend
to zero. In this limit, where the velocity v is attained nearly instan-
AT =
taneously,
we thus
A Tl
get from (152), (153), and (159)
//
2A 7
Ar 2
i.e.
as
Ar 2
2rl
2&"TJ(l-v*[c
(160)
),
Ar x V(l-^ 2 /c 2 )
(161)
we should have. The moving clock (72 is lagging behind the stationary
Cv
Further, in the limit g ->
the two clocks is simply given by
clock
oo,
the
maximum
distance
between
= vAT.
(162)
We shall now see that the same result is obtained if the whole process
treated in a rigid system of reference *S 2 with coordinates (x,y, z, t)
which follows the motion of C*2 in such a way that C2 is permanently
situated at the origin. In the time intervals where S2 is accelerated relative to J&! or to the distant stars, we have a gravitational field in 52
During the time interval < t <. r 2 of magnitude A' = r 2 the gravitais
described by the scalar potential (139). In the interval
r 2 we have x
T'Z
t
an(i i n ^he
T2+ T 2 f magnitude A"/
"
w
interval ri+Vj
/
r2
A'^ we
r^+rl+r^ of magnitude A == r 2
tional field
is
< <
< <
have x
{7^(1
(7#/2c
).
During the
first
period
A'tf
falling freely in the direction of the negative x-axis in
the clock Cj
is
accordance with
VIII,
THE GENERAL THEORY OF RELATIVITY
98
the equation of motion (149). In the period A" it
with the velocity -~v and, finally, in the period A'"
is
261
moving uniformly
it is
brought to rest
Since the systems Sl and S2
moment, the maximum distance
at a point with the coordinates a*
are at rest relative to each other at that
1.
between the two clocks is the same when measured in S2 or in Slf After this
moment
the clock
returns to
the origin with reversed motion,
at rest at the origin of S2 during the whole process, since
the gravitational field is counterbalanced by the external force F.
The
clock
C*2 is
The increase in the proper time of the clock C can now be calculated
by means of the general formula (99) with the expressions for x given
above. In (149), (150), and (151) we have given the solution of the
equations of motion (147). If r\, TJ, and r denote the increase of the
proper time of Cl in the intervals A', A '7, A "7, respectively, we have
l
An =
obviously
2( T ;+rI+TD.
Similarly, since C2 is at rest at the origin x
equal to zero, we have
AT,
+ A"H-A'"0 -
2(A'J
(163)
where x
0,
2(2A'H-A"/)
and
= A7 =
r2
we can
A% the clock (7
X is
A"/V(1~W
2
)
on account of
-(/, XQ
rr
1
~->
"
although
may
2
)
A"T(l-v
/c
(166)
be obtained by putting y equal to
r'2
in (151).
= (MtaBh^ =
c
\g^c)
Hence
(MH
\f7
(167)
c/c
(158').
In the limit as g
A'"
moving with constant velocity
r^(l -v*Ic
(159). Finally, r
r
A"7
-/, and t
on account of
(165)
Thus we get
in a field-free space.
-~
get r[ from
(158').
During the interval
T\
(164)
Hence
on account of
constantly
2(2r'2 -f r 2').
Since C\ starts from the origin with zero velocity,
(151) by putting X Q
*s
r2
This surprising result
oo with constant v
r 2 ->
is
we
in this limit,
get from (165) r\ -> 0. But
the finite limit
rj' approaches
due to the influence on the rate of the clock Cl
= $x ( 1 J7#/2c 2 ) which becomes
of the gravitational scalar potential x
infinite in the limit g->ao.
THE FOUNDATIONS OF
262
Thus, in the limit g ->
and
we
oo,
VIII,
98
get from (163), (166), (168), (162), (164),
(159),
Ar
2r
(169)
the same result as in (160). This result which represents the solution
of the clock paradox is, of course, not surprising, since the proper time is
an invariant which has the same value in any system of coordinates.
i.e.
We
another much simpler example of the
which also the importance of the gravitational
vector potential for the rate of moving clocks is illustrated. Consider a
clock (72 which under the influence of a central force F performs a uniform
shall
now
finally consider
same phenomenon
in
motion in a system of inertia AS\. If the radius of the circle is R
and the constant angular velocity o> p the velocity of the particle is Rw^
and the increase in the proper time r 2 during a revolution is
circular
r2
= 7V(l-# to
2
/c
o>
J(l-R*a>*lc*)
(170)
according to the formula (II. ?J8) The corresponding increase in the
proper time of a clock C\ at rest at the periphery of the circle is
= T=
rl
27T
~".
(171)
CO
Let us now treat the same phenomenon from the point of view of an
observer on the rotating disk of 90. In this system # 2 the clock (72 is at
rest at the point (r
R, & ~~ 0), say, while the clock Cl is rotating with
the angular velocity
'
yn
/
It. (\ is
falling freely under the influence of the
gravitational field with the potentials (74) and (97),
in a circle of radius
(in)
It
is
easily seen that
>
constant, d&/dt
is
a solution of the
clock C2 remains
equations of motion (86) with g tfc given by (73).
at rest, the gravitational acceleration (96) being counter-balanced by the
The
force F.
VIII,
THE GENERAL THEORY OF RELATIVITY
98
263
For the increase r 2 of the proper time of C2 during the time
get at once from the formula (100) for a clock at rest
ZTTJOJ
we now
VU + 2*/c 2 =
r2
CO
CO
Va-r^/c*)
(173)
in accordance with (170).
To determine
we
the corresponding increase of the proper time of C\
(98). Since
have to use the general formula
we
get from (74)
da 2
dx dx
l
Further, using (172),
Hence, from
TI
(98),
27r
f(V(l---r
aj /c
*****
a
)
*****
j^
27T
-
(174)
Thus, in this case, the effects of the gravitational potentials and of the
velocity of the clock C\ on the proper time r l as expressed in the formula
(98) just cancel
and
for TJ
and
r2
we
get the
same values
as before.
The physical interpretation of the effect is, however, completely different in the two cases Tn S1 the effect is ascribed to the velocity of
the particles only, while in S2 the phenomenon is explained as a joint
effect of the gravitational field and the motion.
IX
PERMANENT GRAVITATIONAL FIELDS.
TENSOR CALCULUS IN A GENERAL
RIEMANNIAN SPACE
Four-dimensional formulation of the general principle of
relativity and of the principle of equivalence
IN the preceding chapter we have considered the case of gravitational
99.
which could be transformed away by the introduction of the
pseudo- Cartesian coordinates of the system of inertia / from which we
started in 88. We have seen that the action of the gravitational field
in an arbitrary system of coordinates (x ) is described by the metric
tensor g lk which determines the line element in space-time by the
fields
ec* uation
(1)
The non-permanent
thus characterized by the
property that the interval can be brought into the form (VIII. 40)
gravitational fields are
ds*
-G
lk
dX*dX k
(2)
for
by a suitable choice of the space-time coordinates
or, in other words, the space is a flat pseudo-Euclidean space in this case.
According to the principle of equivalence there should, however, be
all
points in 4-space
no essential difference between permanent and non-permanent fields,
both types of fields satisfying the same fundamental laws. We shall
therefore assume that the gravitational fields produced by the presence
of large masses
that of the earth or the sun, are described
in 4-space in the same way as in the case of the
as, for instance,
by the metric tensor
y tk
produced non-permanent fields. In particular, it is assumed
that the time tracks of a free (i.e. freely falling) particle and of a light ray
artificially
traversing permanent gravitational fields are geodesic lines in 4-space,
given by the same equations (VIII. 86) and (VIII. 87) as in the case of
non-permanent fields. The only difference will then be that the permanent fields cannot be removed completely by a suitable transformation of the space-time coordinates, i.e. ds 2 cannot be brought into the
form
simultaneously for all points in 4-space. Hence, in this case, the
a 'curved' space with a general Riemannian geometry.
4-space
As we shall see in 104, it is, however, always possible in an infinite
number of ways to introduce a so-called geodesic system of coordinates
(2)
is
for
which the
first
derivatives d
lk ldS?
of the metric tensor are zero
IX,
PERMANENT GRAVITATIONAL FIELDS
99
266
and the values of
are equal to Otk at a given point in 4-space. Geothis
means
that the space may be treated as flat in an infinitesimetrically
mal region around every point, in analogy to the two-dimensional case
lfc
where a curved surface may be replaced by the tangential plane in a small
region around the point considered. Systems of space -time coordinates
(#') with the above-mentioned properties may also be called 'local
systems of inertia* for in the case of permanent gravitational fields they
play locally the same role as the systems of inertia in the case of non;
For an arbitrary choice of space-time coordinates
an
(x ),
arbitrary numbering of the events in physical space,
the quantities g lk can therefore always be experimentally determined
by the same methods as those described in 89.
permanent
l
i.e.
fields.
for
According to the general principle of relativity, the laws of nature
expressible in the form of equations which are form-
must now be
"
invariant. Thus, if the law in question
is
expressed by equations of the
dA dB
B '""''"'
} =0
-
'
B,... are physical quantities, we have in another arbitrary
of
coordinates (x >l ) the same functional relation between the
system
where
physical quantities in
(x'
i.e.
The only difference from the case considered in 35 is that now the gravitational quantities g lk will be among the set of physical quantities
A, J5,... entering into the equations (3).
In the special theory of relativity, the covariance of the laws of nature
under Lorentz transformations could be very elegantly expressed by
means of the four-dimensional tensor calculus. To obtain a similar
representation of the laws of nature in the general theory of relativity
we shall have to make a generalization of the tensor calculus developed
in
Chapter IV for pseudo- Cartesian systems of coordinates.
In the case of non-permanent gravitational fields, this generalization
simply consists in the rather
trivial
extension of the notion of vectors
and tensors to general curvilinear space-time coordinates, the geometry
of 4-space being the same as in the special theory of relativity. In the
general case of permanent fields, however, the structure of the 4-space
itself is different and the development of a tensor calculus for a general
Riemannian space will be required. Formally the difference between
these two cases
is
not very great and, as
we
shall see,
most of the tensor
PERMANENT GRAVITATIONAL FIELDS
266
IX,
99
relations holding for curvilinear coordinates in a flat space can be used
also in a general curved space.
and covariant components
100. Contravariant
of a four -vector
be an arbitrary system of curvilinear coordinates in 4-space.
this space is then completely defined if the components
tensor
are given functions of the space-time coordinates.
of
the
metric
fj ik
Let
(x
The geometry in
We
assume that the (j lk satisfy the conditions (VIII.
event point This means that the determinant
==
is
52) at every
shall
negative. Let
012
013
014
031
032
033
034
041
042
043
044
(4)
A lk be the conjugate minor of the element g lk in the ith
row and kih column.
determinants
I0i*l
011
we then
From well-known theorems
of the theory of
get
summation over
(no
i)
\.
(5)
and
i
Defining now a symmetrical scheme
g*
we have, according
to
of quantities g tk
by
- d* = / S
(6)
(5),
= ?* = 8?,
=*
f f
8* = (
1
0,/ff*
where
If
we introduce a new system
for
of coordinates Jby the transformation
(9)
we
get, exactly as in the case of a flat space considered in
tions VIII. 54-58),
1
- 4 dx" =
=
(8)
-^ k
x' 1 =- r''(j<?)
'
(7)
a'k dx' k
88 (see equa-
dx"
dx' k
(11)
g\ k
9tk
=
=
&\s$glm
<*(<*kSi m
IX,
PERMANENT GRAVITATIONAL FIELDS
100
In virtue of the transformations
is
(10), (12)
the expression for the interval
^ _ ^ ^^ = ^ ^ w
invariant
267
*.
13)
always possible to choose the transformation (9) so as to make
g lk equal to the G lk defined by (VIII. 41) at a given point in 4-space. In
general it will, however, not be possible tp obtain this result simulIt
is
taneously for
all
must
since they
points; for the coefficients aj, cannot be chosen freely
satisfy the integrability conditions
following from the definition of the aj. in (10). This will be possible only
for a flat space, in which case the functions g lk satisfy special conditions
(see
107).
A vector at a definite point
is now defined as a quantity which has
)
four components a in every system of coordinates satisfying the same
transformation law as the coordinate differentials in (10), i.e.
l
(x
a' 1
On account
==-
4a
fc
(15)
of (11) the reciprocal relations are
= &k a' k
(15')
Since the coefficients a k and & k are functions of the coordinates (a: ),
the transformation laws (15) and (15') have a definite meaning only
with reference to the particular point with which the vector a is con1
nected.
In curvilinear systems of coordinates we have to distinguish between
the contravanant components with the transformation equations ( 1 5) and
the covanant components a l of the same vector, defined
in
every system of coordinates.
by
the equa-
On account of (7) the relations reciprocal
a *__*
9 a k-
to (16) are
The operations
(16)
and
(17) are called lowering
m\
l{
(
and
raising of indices.
In a Cartesian system of coordinates in Euclidean space we have
and there
will
be no difference between the covariant and the contra-
variant components.
PERMANENT GRAVITATIONAL FIELDS
268
From
(12), (15),
and
(11)
100
IX,
we get the following transformation equa-
tions for the co variant components:
or
the reciprocals of which are
we can use pseudo-Cartesian coordinates and
the connexion between the covariant and the contravariant comIn a
flat (3 -fl) -space
ponents of a \ector in this system of coordinates
or
is
simply
~
,
i.e.
in this case
O = #
tk
we have
(20')
?fc
and (VIII. 41). If we had used the real representation of vectors in the special theory of relativity, it would have been
necessary to distinguish between covariant and contravariant comin accordance with (7)
ponents
in
Chapter IV.
It
was
just in order to avoid this slight
com-
plication that the imaginary time components were introduced by
which the 4-space was made fornially Euclidean. In any case, it is easy
to pass over from the imaginary to the real representation. It is only to
be remembered that the contravariant components of a vector in the
real representation are obtained from the components in the imaginary
representation by dropping the symbol i in the fourth component.
covariant components of the vector are then obtained from (20).
From
this rule it follows that the
norm
The
of a vector, which in the
imaginary representation was defined by (IV.
tion takes the form
2 __
25), in
the real representa/oi\
and, since this expression is invariant for all coordinate transformations, (21) will hold also for curvilinear coordinates. In fact, we get
from
(18), (15),
and
(11)
1
a\ a'
We may
dt{a; ft
0,a
therefore take (21) as the
Biemannian space. According to
different forms
^=
Sj rt
norm
16)
a,a
a, a'-
22 )
of a vector also in a general
(21 ) may be written in the
and (17),
^ _ ^^ = ^^
Similarly, the vector product of
invariant
two vectors a and
(23)
b is given
by the
IX,
PERMANENT GRAVITATIONAL FIELDS
101
Tensor algebra
The generalization of the tensor
269
101.
calculus developed in
39-44 for
Cartesian systems of coordinates to the general curvilinear coordinates
of Riemannian space is now obvious. A tensor of rank n in 4-space is a
n
quantity with 4 components which transform with respect to each
index like a vector,
is given by (15) or (18),
contravariant or covariant with respect to
the transformation
i.e.
according as the tensor
is
the index in question. The connexions between the covariant and
contravariant components of a tensor are given by the general rules
and (17) for lowering and raising indices.
The transformation laws for the contravariant and covariant compo-
(16)
nents of a tensor of rank 2 are thus
=
4=
*"*
respectively,
ajc&#*
(25)
&\&Wim
(26)
>
and the connexions between the covariant and contra-
variant components are
(27)
'in
every system of coordinates.
On
account of (11) and
(12), the equaBesides the purely
can also form the mixed
tions (25-27) are easily seen to be compatible.
contravariant and covariant Components
we
components
tf
= 0% =
0,i*
t\^gtik ^g
and
ki
vl
>
(28)
which transform according to the laws
In general, t\
All
will
symmetry
be different from
properties like
iik
are invariant properties.
__
They may
_^ t
ki
(30)
also be expressed as
which, on account of (27), (28), are equivalent to (30). Thus the mixed
k
components t* and t i of a symmetric tensor are equal and may therefore
k
simply be written as t
.
A comparison of (12) and (26) shows that the quantities gik themselves
are the covariant components of a symmetrical tensor of rank 2 the
metric tensor. Further, we see from (7) that the mixed components g* of
PERMANENT GRAVITATIONAL FIELDS
270
101
IX,
this tensor are given by the Kronecker symbol 8f and that the contravariant components of the metric tensor are equal to the quantities
lk
g defined by (6).
41,
Similarly, as in the case of Cartesian coordinates considered in
also in the general case form new tensors by the processes
we can now
of addition, direct multipli cation, and contraction. By addition of two
we get a new tensor of rank n, and by direct multiplica-
tensors of rank n
tion of
It
two tensors of the ranks n and
m we get a tensor of rank n-\-m.
should be remarked, however, that these processes have an unam-
biguous meaning in the general case only if the two tensors belong to
the same point in 4-space. Finally, the process of contraction which in
curvilinear systems of coordinates consists in equating
an upper and a
lower index, and summing, reduces the rank of a tensor by 2. By contraction of a tensor of rank 2 we thus get a tensor of rank 0, i.e. an
invariant. From the transformation law (29) we see at once that
is
an invariant which, by means of
forms
/
l
/T
(
Jik
*M
l
(28),
be written in the different
may
n^t
l
(/
ik
f
l
(w\
V)
k-
An example
of a combined application of the processes of
and
contraction is offered by the equation (24).
multiplication
direct
Pseudo- tensors. Dual tensors
= af and a = |a* be the determinants corresponding to the
scheme of transformation coefficients cxf and af respectively. According
to the multiplication rules for determinants, we get at once from (11)
102.
Let a
.=!,
and from
(12)
where
and
|a|
g'
|a|
\g( k
=
\
dc.g.ci
(34)
& 2g
---,
are the absolute values of the determinants a
and
a.
now
defined as a quantity whose components
pseudo-tensor
transform like the components of a tensor, except that they are multiplied
by the
is
sign
|a|
of the transformation determinants.
If
ex
and
|a|
consequently a are positive, a pseudo-tensor therefore transforms like
a tensor of the same rank*. In the same way as in 43, it is now easily seen
IX,
PERMANENT GRAVITATIONAL FIELDS
102
that a quantity
A lktm
whose components
in every
271
system of coordinates
are equal to the Levi-Civita symbol 8 lWm transforms according to the law
,
A'tklm
From
(35)
and
we thus
(36)
otflSt&lc&A^.
(36)
see that the quantities
are the covariant components of a completely antisymmetric pseudo-
tensor of rank
4.
way as in 44, we can now to an antisymmetrical tensor
of rank n adjoin a dual pseudo-tensor of rank 4 n by means of the
lk
pseudo-tensor (37). Thus, if F are the contravariant components of an
In the same
antisymmetrical tensor, the covariant components of the dual tensor
are given
by
^^
y_
Fm =
,
^^
g)
(sg)
i.e.
3t
"
'
Two
infinitesimal vectors a 1
b k define
(39)
a parallelogram described by
an antisymmetrical tensor with the contravariant components
O ik
al b k
the area a being given by or
\o lk a
The corresponding dual tensor
is
?Ar
orthogonal to a
ak b l
(40)
lk
.
i.e.
Three infinitesimal vectors a 1 b k d define a three-dimensional parallele,
piped described by the antisymmetrical tensor
a1
bl
c'
bk
ck
'
or
by
its
(42)
dual pseudo-vector
(43)
which
is
l
orthogonal to the vectors a\ b
^a
The volume V of the
==
Vb
%
parallelepiped
V,c
is
i.e.
0.
given by
p=_F P=-ii
l
ftll
F*.
(44)
PERMANENT GRAVITATIONAL FIELDS
272
dm define a four-dimen-
k cl
>
Finally, four infinitesimal vectors a\ b
sional parallelepiped described by the tensor
,
Cl
d*
a*
ck
dk
a
am
cl
d*
cm
flm
;<
bm
102
IX,
(45)
by the dual pseudo-invariant
or
V.Wm
V(-flO'
4'
4!
.e.
v<-
V(-P) a
Jf a z
61
(46)
d3
4
d are infinitesimal vectors lying in the directions of the
coordinate curves and of lengths dx l dx 2 dot?, dx*, respectively, we have
',
a1
(da:
l
0,0,0), b
(0,dx
dimensional volume element
dZ
This
is
and the corresponding fourthe
given by
pseudo-invariant
0, 0), etc.,
is
V(-flO dx*dx*dx?<fa*.
(47)
the generalization of the expression (VIII. 17) for the volume
element in a two-dimensional space with positive definite metric. In a
3-space we can similarly adjoin a pseudo-tensor of rank 3 n to an
antisymmetrical tensor of rank n.
(See
103. Geodesic lines. Christoffel's
The geodesic
lines are defined
dX
gik
ik
Appendix
5.)
formulae
by the equations
(VIII. 30)
~~
(48)
dA
~dX
where A is an arbitrary invariant parameter and the indices
from 1 to 4. (48) may also be written
ik
mi
Multiplying this equation by g
i,
k,
run
fyu\ dxk d^ _,
we
get,
on account of
(7),
'_^W\^ ^ _
*
or
(49)
PERMANENT GRAVITATIONAL FIELDS
103
IX,
273
with
The
Tlkl
quantities Fjj,
symbols.
defined
They obviously
F
by
F'
(50) are the Christoffel three-index
satisfy the relations
--^
4-F
(51)
The connexion between F^ and F w is the same as that between the covariant and contravariant components of a tensor but, as we shall see,
t
the Christoffel symbols do not transform like a tensor.
Since the equations (48) or (49) are the Euler equations corresponding
to the invariant variational principle (VIII. 21, 29), they must hold in
every system of coordinates.
we
By
differentiation of the equations (10)
therefore get
dx
_ ^ dx'
d\
~
dX
U>
'
w|
(JL
GOL^ CLX
1
ic,'
where we have used the equations
Using (52) in (49) we get
(J/X
dX
dX
(49) in the
/r\^\
system of coordinates
^ -'diA--
(x'
==0
and, since this equation must hold for independent values of the variables
dx' k /dX, the expression inside the brackets, which is symmetrical in k and
If we multiply the relation obtained in this way by a
I, must be zero.
we
get,
on account of
(11),
the Christoffel formulae
(53)
where the coefficients d, a are
term on the right-hand side of (53) is zero and the
Christoffel symbols transform like the components of a tensor. For more
are
general transformations this will not be the case, and the Fj^, Tikl
By
linear (affine) transformations,
constant, the
first
therefore called affine tensors.
If the g lk are constant, as in the case of the pseudo- Cartesian system of
coordinates in a flat space, the Christoffel symbols (50) vanish.
3695.60
PERMANENT GRAVITATIONAL FIELDS
274
104. Local
systems
IX,
104
of inertia
In general, it is not possible to introduce a system of coordinates which
makes the components of the metric tensor independent of the coordinates, but, as
we
approximately true
4-space.
in
now, we can always ensure that this is
the immediate surroundings of a given point P in
shall see
More exactly, we can always
a geodesic system
for
which
()g lk /dx
system of coordinates ($J
at the point P. Let (x ) be the
let T'kl (P) be the values of the
find a
original system of coordinates, and
If
('hnstoffel symbols at the point P.
P denote the coordinates of
(x'
this point, the transformation
x'
r
8
r
-x^+\r^(P)(* -~x >)(x*--~x 1 ,)
l
will lead to the desired result. If we identify the
the system (x
we
P we
At the point
get from
(54)
primed system in (53) with
(54)
thus have
&UP)
--
(P)
?|
X
i,
= - r;,(m &;,
(55)
Cj
and
if
we use
(55) in (53)
we
get
l"'u(P)
(56)
at the point P. From (50) and (51) we then find that also Tltkl and
l
are zero at this point. Further, the components of the metric
i jdx
tensor y lk at
are, on account of (55),
and the derivatives of the gravitational potentials y and % defined by
(VIII. 63, 94) are zero. Thus the gravitational acceleration is zero at P,
l
the gravitational field has been locally transformed away. The system
of reference R corresponding to the coordinates x l is a local system of
inertia.
o
of the system R relative to the original
with coordinates (x ) is obtained from (54) by putting
0,
The motion of the
system R
which gives
origin
O
1
- jr*-a+
&=
PERMANENT GRAVITATIONAL FIELDS
104
IX,
275
By differentiation of this equation with respect to an arbitrary parameter
A
we
get
At the time
coordinates of
m-
r
dw + rs(
,
'
dX~
Zp/c, corresponding to the event P, the space-time
are
and
x'r
O
At
this
moment
the motion of
O is thus given by the equations
2l
0.
rr
8
tJjr
(58)
i) -
which are the equations of motion of a freely falling particle v/hich is
momentarily at rest relative to H (cf. (49)).
Any geodesic line which goes through the point jP, including the time
track of freely falling particles and light rays, is in the system # described
by the equations (49) with the Christoffel symbol given by (56) Hence
l
we
get simply
d *
dX
0.
(59)
In a small region around P the local system of inertia thus has the same
properties as a usual system of inertia. Without any change in the
system of reference we may now further introduce local pseudo- Cartesian
coordinates
X m
1
the metric tensor
R, by a transformation of the type (VIIT. 59), so that
G lk
in this
new system
for
for
for
(1
--
is
fc=- 1,2,3
(60)
at the point P. The necessary transformation is even linear, therefore
the derivatives of the metric tensor will remain zero at P, i.e.
o
8
'
(P)
0.
(61)
dx-
In the following we shall always use such space-time coordinates in
the local systems of inertia. By Lorentz transformations of the variables
will
we can pass over
to new local systems of inertia which in general
be moving relative to the original one.
PERMANENT GRAVITATIONAL FIELDS
276
104
IX,
o
1
can be
In a small region around P, where terms of second order in
neglected, the metric tensor may be regarded as constant in these
systems.
In accordance with the principle of equivalence, it is now
all laws of nature at the point P have the same form as in the
assumed that
when expressed in terms of
the local pseudoa
transformation
of coordinates
By simple
then obtain these laws in a general covariant form. This requires
special theory of relativity
Cartesian coordinates
we may
the development of tensor analysis in a general Riemannian space.
105. Parallel
displacement of vectors
Let a be the contravariant components of a vector at the point (x l ).
By means of the Christotfel formulae it is now easily seen that the
1
quantities a*'
_ a
a with
l
-\~d if
dt) a
= -r
kl
dx k a
(62)
transform like the contravariant components of a vector at the neighbouring point (x* -\-djr ). From (53), (10), and (11) we obtain
1
(63)
Further,
we
get
by
differentiation of (11)
?*!
ox n
and the
first
term
dJ>n a'
if
we neglect terms
(in
v
dx">
in the brackets
where we have made use of
Hence (63) becomes
and
4+^ =
(1 1)
Pa* l
^Qx m
'-
may
and
therefore be written
(14).-
da
dx m a n +ot r p
l
r
,
>
of second order in dx l the transformation law for
the quantities a +rf^a can be written
1
=
where
* lH (x+dx)(a*+dp a)
(64)
n (x-{~dx) are the transformation coefficients taken at the point
l
1
1
a l +d p a i are the contra(x +dx ). (64) shows that the quantities a*
variant components of a vector at the point (x^+dx 1 ). In a flat space,
<x,'
IX,
PERMANENT GRAVITATIONAL FIELDS
105
277
is identical with the vector obtained by parallel
displacement of the vector a from the point (x to the neighbouring point
i
for if we introduce pseudo- Cartesian coordinates, the Christoffel
(x -\-dx
and
therefore also dp a 1 vanish and the components a* 1 and a 1
symbols
of the two vectors are equal in this system. If we use a curvilinear
the vector a* 1
system of coordinates in a
the two vectors
flat space,
will, thus, differ
the contravariant components of
amount dlt a l defined by (62).
by the
now
natural also in a general Riemannian space to define the
parallel displacement of a vector by the equation (62).
i
If
is a local system of inertia for the point considered, the comIt is
ponents a of a vector at this point are unchanged by parallel displacements exactly as in a pseudo- Cartesian system of coordinates in a
1
pseudo-Euclidean space.
From
(62) it follows that the
1
product of two vectors a
and
parallel displacement, for
from
norm
of a vector,
and
b l at the point (x l ), are
(62)
we
also the scalar
unchanged by a
get
(65)
on account of the identity
(66)
^f-flV^-^rj^O
following from (50) and (51).
Using one of the other forms (24) in which the scalar product can be
written,
we
get
dp (a b
t
dp a l .b l
a l Y kl dxk b
l
Since this equation must hold for an arbitrary vector 6', we get for the
change in the covariant components of a vector by a parallel displace-
ment
d,a t
= TU dx*a, =
r,. lfc
dxW.
Finally, expressing the scalar product in the
means of
(67)
= <7'*a)
(67)
lk
form g a
b kt
we
get
by
PERMANENT GRAVITATIONAL FIELDS
278
IX,
105
and, since this equation must hold for arbitrary dx\ a iy 6 t the equation
,
0~1 If
\
must be
==
(68)
identically satisfied.
Multiplying (66) by g
lk
and applying
(7),
we obtain
Farther we get by means of a well-known theorem the derivative of the
in the form
determinant </
|j/ (fr
-
where we have made use of
Hence
(6)
and
(7).
r-
11
a relation which we shall use
(69)
'
later.
Consider again a geodesic line defined by the equation
an invariant parameter,
,
If A
(49).
is
a four-vector lying in the direction of the tangent.
(49) may then be written
1Jl
is
The equation
A comparison of (71) with
(62)
the geodesic line are obtained
shows that the different vectors
by
along
parallel displacement along this line, a
property which they have in common with the straight lines in a Euclidean space. The geodesic line connecting two points is thus not only the
line
with a stationary value of length,
The norm of the vector
U Uk
l
it is
also the 'straightest' line.
must therefore be constant along the
line,
independent of A, in accordance with (VIII. 31).
If this property of the geodesic lines is expressed in terms of the
covariant components t^ of the vector dx l /dX, we get, according to
i.e. g^ k
(67)
and
is
(51),
*% =
This equation
is
rllk
uw = Krw +rM)iw' = ~ ^ v
u*.
(72)
identical with the equation (48).
'
The equations (62) define the change in the components a 1 of a vector
by an infinitesimal parallel displacement along the vector (dx ). The
l
PERMANENT GRAVITATIONAL FIELDS
105
IX,
279
change of a by a displacement along a finite curve may thus be
obtained by integration. In aflat space the total change of a 1 by parallel
displacement along a closed curve will be zero. This is seen at once if
1
total
we
this
use a Cartesian or pseudo- Cartesian system of coordinates, for in
1
system the components a are not changed at all by the displace-
The
ment.
closed curve
is
also be true if
vector a*
obtained by the displacement along the
thus equal to the original vector a and this must then
we afterwards introduce curvilinear coordinates. In a
final
curved space, however, the final vector a* will in general be different
from a the difference a* ~a depending on the closed curve (see 107).
1
given a parallel displacement from a point Pl to a
point 1 g along a certain curve connecting I\ and P2 the resulting vector
a* will depend on the form of this curve if the space is curved, while it is
Thus,
if
a vector
is
independent of the curve in a flat space. This
difference between curved and flat space.
106.
As
is
in fact the only essential
Tensor analysis. Covariant differentiation
in the case of the flat
of relativity,
we speak of a
pseudo-Euclidean space of the special theory
tensor (or pseudo-tensor) field of rank n in a
general Ricmannian space
if
a tensor (or pseudo-tensor) of rank n
is
connected with every point in this space. Similarly as in 48, we can
derive from such a field a new tensor field of rank n-\- 1 by a differentiation process.
From
a tensor
field
of rank
f (*') =
we can thus
derive a vector
field
grad
grad,*
By means
of (73) and (10)
which shows that the
we
d<f)/dx
0, i.e.
from a scalar
^(x),
</>
field
(73)
with the co variant components
g.
(74)
get at once
are in fact the covariant
components of a
vector.
However, if we try in the same way to form a tensor field of rank 2
from a vector field a 1 by differentiation of the transformation equations
v
'
,76,
k
/3x will be the mixed components
of a tensor only if the coefficients aj are constant. On the other hand,
From this equation we see that the da
PERMANENT GRAVITATIONAL FIELDS
280
we get by means of the Christoffel formulae (53) and the relations
and (IT)
106
IX,
(
1 1 ), (
14) ,
(p. 276)
(77)
By
we
addition of (76) and (77)
find that the quantities
a'^g+IlX
(78)
transform according to the law
Thus, by the process of 'covariant differentiation' (78) of the contravariant components a of a vector w e get the mixed components of a
r
tensor field of rank
2.
This process may be described geometrically in the following way.
Let a (P) and a (P') be the vectors of the vector field connected with
the neighbouring points P and P' with coordinates (x and (x l -\-dx l ),
l
The
respectively.
differences
da 1
a*(P')-a*(P)
f^*
dx k
then not be the components of a vector, since a'(P) and a'(P')
belong to different points. However, if a*'(P') denotes the vector
will
obtained by parallel displacement of a 1 from
a (P')-a* (P')
l
to P', the difference
a (P )~c
will represent an infinitesimal vector at the point P'. Neglecting terms
of the second order in (dx k ), a k dd* will also be a vector at P, and since
l
must hold for arbitrary infinitesimal vectors dx the quantities
must be the mixed components of a tensor of rank 2.
1
this
If this consideration
vector
field,
we
is
a\ k
applied to the covariant components a of the
the formulae (67) instead of (62) for the
t
find, using
parallel displacement, that the quantities
a '-
fc
"~
'
jia '
(79)
components of a tensor. Tn a geodesic system and, in
particular, in a Cartesian system of coordinates in a flat space, where
the Christoffel symbols are zero, the covariant differentiations (78) and
are the covariant
(79) are ordinary differentiations.
IX,
PERMANENT
106
CSRA\
JTATIONAL FIELDS
281
Let us now consider the special case, where we have only a vector
a l (A) connected with each point on a cm ve with the parametric representation
z*=
'(A).
We can then define the covariant derivative of this vector with respect to
The quantity
Do'-
d\
where
llin
P'-+P
obviously a vector at the point
for from (62) we see that
_Da*(A)/dA
coordinates #'(A);
rfA
is
[o'(f')-------
AA
with the coordinates
approaching the point
with the
Jllll
AA
i"-*p
(80')
a;
(A+AA)
in this limiting process is
along the curve. Similarly
variant derivative of the covariant components a
we
get for the co-
''-'
By means
of (51)
we
get
DaJdX and Da'jdX are the covariant and contravariant components
of the same vector.
i.e.
According to (71) and (72) the equations for the time track of a free
particle can now also be written
or
d\
d\
The process of covariant differentiation may be applied also to
tensor fields of higher rank. ( Consider, for instance, a tensor field of rank
2 with contravariant
k
components V
Since each index
is
transformed
PERMANENT GRAVITATIONAL FIELDS
282
separately according to the same rule as for a vector,
quantities
it is
106
IX,
clear that the
^. tfc
with two contravanant indices and one covariant index, are the mixed
components of a tensor of rank 3. This may be verified in the same way
by means of the transformation laws of tensors and
as for a vector field
Christoffel symbols.
*W
seen that
it is
Similarly
pr
l
t>k
pr
iv
i
/
t
<MrA,
T-J
AM<r
(82)
f/<
k L
-TTj 'T
are purely covariant
of rank 3.
These rules
may
ir
]i)
V
t
lr
k~
kl 'r
and mixed components,
respectively, of a tensor
be extended to the case of a tensor
number of terms containing
and if we put
the
Christoffel
field
of rank n,
symbols being n in
this case,
<
Ytlt
-*
dx
(83)
in the case of a scalar field, this rule also applies to tensor fields of
Since the scalar product aj) 1 of two vector fields a
is
rank 0.
a scalar
field,
we have
(M)
'.*,
i.e.
the usual rule for differentiation of a product holds also for covariant
This rule is easily seen to hold for the contracted
differentiation.
product of any two tensors of arbitrary rank
lk
(t
The
identities (66)
ak )
and
(68)
g lkjt
i.e.
lk
tl
tl
a k +V k a kil
may now
0,
for instance
be written
g*j
0,
(86)
the covariant derivatives of the metric tensor are zero.
differentiation of (16)
"i,*
and
k
9ik,i<*>
(17)
we
+g lk a k tl
we have
By
covariant
therefore get
g lk a\i,
a',
g*a kjt
(87)
l
are components of the same tensor of
i.e. the quantities a lk and a
fk
rank 2. In the same way we see that the quantities t ikj, Pk j, and t lkl>
defined by (81) and (82), are components of the same tensor of rank
3.
PERMANENT GRAVITATIONAL FIELDS
106
IX,
283
To obtain a generalization of the differential operators defined in
48 for the special case of pseudo-Euclidean space, we simply have to
48 by covariant differentiations. For
replace the differentiations in
the curl of a vector
al
field
we thus
get
since the terms containing the Christoffel symbols cancel.
The covariant expression for the divergence of* a vector
by
contraction of the tensor a1 fc
}==a<
On
account of (69) this
may
also
fl
(89)
be written
In the (3+1) -space (/ is negative, i.e.
and \g\
definite metric tensor g
\g\
>
is
obtained
+r< r ar.
** = s+JBi s> d'Alembert's operator
is
i.e.
g.
561
</,
www
""
for a space with positive
The covariant expression
for
now
(91)
The contra variant components of the divergence of a tensor field
(92a)
'
and
similarly
we
T tk are
get for the covariant components
div ( {Tt
*}
= T** =
^L^(\g\)Tf)-r
For a symmetrical tensor
r, ta
T'<.
(926)
this reduces to
T"
on account of (51).
For an antisymmetrical tensor
F lk
the last term in (92 a)
(92c)
is
zero on
account of the symmetry of Fj^ in the lower indices; hence
(93)
PERMANENT GRAVITATIONAL FIELDS
284
we
Further,
106
IX,
get for the curl of an antisymmetrical tensor
(94)
since the terms containing the Christoffel symbols cancel.
The
generalization of Gauss's theorem (IV. 191)
(43), (47),
and
(90),
we get
an arbitrary vector
for
is
obvious.
field
a 1 in the
Using
(3
+ 1)-
space
f div{a}
dZ
--=
,f
(V(-(7)a
a' dV> =-
J a'V(-ff)
where (dx k ),
(A#
(8xf),
WI
107.
dimensional boundary
8 ljHwl
rf^S^A^,
(95)
are three infinitesimal vectors in the three-
of the four-dimensional region.
The curvature tensor
Let a k be an arbitrary vector
field
by covariant differentiation. It
and
may
the tensor of rank 2 obtained
be written in the two forms
l\^
(96)
Similarly the tensor of rank 3 obtained by a further covariant differentiation can be written in the two forms
n
a
k,lm
where ar
is
;Z
^a k l
t
*fafi~~~
rr
L
n
km ar,l~~ lr lrn a
k,r
r
obtained from (78) or
~
-
^ ak
~^rn~~
>
r
rr km n r '~~ 1F lm
n
a
k,r*
'
(W\
Jl )
(87).
Using the first expression (96) in the first form of a klm in (97), we
get an expression for a k tlm which is a linear function of a and its first
and second derivatives. However, if we subtract the tensor a k tml obtained
l
by interchanging the order of the covariant
tives of a disappear and we get simply
where the
coefficients of a t are given
pi
Wm
~~
f_fc!
ftr"*
--
km
differentiations, the deriva-
by
pi
^f~
Using instead the second expressions
m prkl ~ ptrl prkm
(96)
and
(97),
/QQ\
'
we get in the same
'
"^,*aS
(100)
PERMANENT GRAVITATIONAL FIELDS
107
IX,
285
where
___
jf
f_2_f.W
v*i,km
pr
pr
Since the left-hand sides of (98) and (100) are the components of a
any vector field a,, the quantities R'um and K lM n must be the
tensor for
components of the same tensor of rank
*'u
4, i.e.
ff'-^H.
(102)
K lUm is called the
Riemann-Cfuistoffel curvature tensor. The
the
commutation
law for the covariant differequations (98) represent
entiation of a vector field. The corresponding law for the covariant
The
tensor
differentiation of a tensor field
lk is
easily seen to be
The geometrical meaning of the curvature tensor becomes apparent
if
one considers a parallel displacement of a vector at along the contour
of an infinitesimal parallelogram defined by two infinitesimal vectors
105, the vector a* resulting from this
(dx ), (8x ). As mentioned in
l
1
process will in general differ from the vector a
displacement laws (62) and (67)
calculations that the differences
Aa
a*
a1
it
may now
Aa
By means of the parallel
be verified by elementary
a* t
ai
between the components of these vectors are given by
Aa<
\K\ lm <* do*",
do2m
where
dtfx m
Aa
dx m
iWlIl
a*
d<fi",
(104)
y*.
In a flat space, where we can introduce a system of coordinates in
which the components of the metric tensor are constant, we have
RMm =
bviOU8ly
space. It
is,
it is
is
possible to find a transformation (9) which
transformed components
/l
(x
(see
(105)
thus a necessary condition for the space to be a flat
however, also a sufficient condition; for if (105) holds at all
This equation
points,
0-
Appendix
6).
g'lfc
makes the
independent of the space-time variables
PERMANENT GRAVITATIONAL FIELDS
286
From
(101)
IX,
107
follows immediately that the curvature tensor satisfies
it
the relations
k +-R lmkl
Q.
(1066)
Besides these algebraic relations the curvature tensor satisfies a 3^t of
differential identities which may be obtained in the following way.|
On account of the 'general rule for covariant differentiation of a contracted product,
we
get from (98)
a k,lmn
a k',mln
== ~~
by covariant
^klnnn a i~~
differentiation
klm
to this equation the two equations obtained
tion of the indices /, m, n, we get
Adding
)i-
by
permuta-
cyclic
Each of the three brackets on the left-hand side may be transformed by
means of the equations (103) applied to the tensors a ktl a k ttn and a kttl
respectively. In this way we get six terms, three of which cancel on
,
account of (1066), while the other three cancel with the terms
second bracket on the right-hand side of (107). Hence we get
Since this equation must hold for an arbitrary vector field a t
we
>
in
the
are led
to the Bianchi identities
On
0.
(108)
account of the identities (106) the number of algebraically indepenis 20 in a four-dimensional
dent components of the curvature tensor
space, 6 in a three-dimensional space,
108.
By
The contracted forms
and
in a two-dimensional space.
of the curvature tensor
contraction of the tensor JR\ ?W of rank 4 we get a tensor of rank 2
may be written in the different forms
which, on account of (106 a),
X*
As the
==
^rk
last expression is
-^fcr
= ~^rk ^
equal to the
contracted curvature tensor
is
first
with
*,*%
i
^*rr
and k interchanged,
further contraction
we
p. 169.
Bergmann,
(no)
get the curvature scalar
*=
t See, e.g., P. G,
this
obviously symmetrical
** = **
By
(109)
*t
Introduction
0**,
to the
(in)
fc
Theory of
Relativity,
New
York, 1942,
IX,
PERMANENT GRAVITATIONAL FIELDS
108
287
Contracting the Bianchi identities (108) with respect to the indices
I, and using the relations (109) and (106 a), we get
and
*m,
+ X'km.>-Bt
Ht
**,- &",- X*,* =
Or
Further contraction with respect to the indices k and
R n -2Bt ul
by multiplication by
m gives
and application of (86)
g
rule for covariant differentiation of products,
or,
0.
(#*- iy'^U =
and of the general
o.
(112)
This equation expresses the fact that the covariant divergence of the
symmetrical tensor
is
On account
zero.
of the
\g*
(113)
symmetry property,
this tensor has ten inde-
pendent components only.
From
for
R lk
(109), (99),
and
(69)
we
get the following explicit expressions
~^
_
~"
fc
llkr
lk
'
'
THE INFLUENCE OF GRAVITATIONAL FIELDS
ON PHYSICAL PHENOMENA
109.
Mechanics
of free particles in the presence of gravitational
fields
BY means
of the formalism of the general tensor calculus and the
assumption made at the end of 104, based on the principle of equivalence, the physical laws of the special theory of relativity can now be
generalized in an unambiguous way. Since the tensor equations of the
special theory are assumed to hold in a local system of inertia, i.e. in a
geodesic system of space-time coordinates, the problem of finding, for
instance, the fundamental equations of mechanics and electrodynamics
in the presence of gravitational fields reduces to a purely geometrical
problem in 4-space.
Let us
first
consider the motion oi a particle in an arbitrary system of
coordinates v(x l '). Let
xl
,/
/i\
(1)
x*(r)
be the equation of the time track of the motion, r being the proper time
of the particle measured by a standard clock following the particle. The
contravariant components of the four- velocity are then, on account of
(VIII. 98),
where U
velocity
di
tf=*
dr
dx^jdt are the
(IV.er),
(2)
contravariant components of the spatial
and
l
f
is
the analogue of the Lorentz factor in the presence of a gravitational
with the dynamical potentials (y t x)- In a local system of inertia,
field
the expressions (2) become equivalent to (IV. 39).
The covariant components of the four- velocity vector are
and (VIII.
by (IX.
16)
63, 64, 94)
i.e.
(4)
X,
ON PHYSICAL PHENOMENA
'109
289
K
y lK U are the covariant components of the three-dimensional
velocity vector, calculated from the contravariant components by
where u
means of the spatial metric tensor y lK By purely spatial transformations
.
x'
x'i(x*)
x'*>
x*
(5)
the spatial parts U and UL of the four-velocity transform like the
contravariant and covariant components, respectively, of a vector; but
unless the system of coordinates (x ) is time-orthogonal, U L and UL
l
will represent different space vectors.
course
which also follows from the
The components of the
[/t
According to (IV. 41) we have of
= _^
(6)
explicit expressions (2)
and
(4) for
and
four-acceleration in curvilinear coordinates
are obviously
A = DU*
r- =
ar
<W*
-~j
ar
MU u
DU
A - ~y
dU
---
>
dr
ar
T
E7*t7'
u
l /lA u
.
(7)
(t)
obtained by covariant differentiation (IX. 80, 80") of U 1 and Ui with
respect to the proper time. In a local system of inertia, (7) reduces to
(IV. 42). According to (IV. 41')
we have
UA =
l
U^A,
0.
(7')
The contravariant and covariant components of the four-momentum
by (IV. 50),
vector are now,
P>
rfe
P -
C/%
7&
19
(8)
is the proper mass of the particle, i.e. the mass measured in a
system of inertia. $ is equal to the rest mass which the particle
would have when placed in a usual system of inertia.
For a free particle, i.e. a particle which is acted upon by gravitational
forces only, we have ih a local system of inertia (x l )
where
local rest
T-* T =
dr
In a general system (x
'>-
dr
these equations take the forms
DP
where
(10a)
dr
dr
represent the covariant derivatives of the
3596.60
(106)
,, tfc
four-momentum
vector.
INFLUENCE OF GRAVITATIONAL FIELDS
200
DP
109
X,
are equal to the differences of
the components of the momentum four-vector Pi (r-\-dr) connected with
the point x l (r-\-dr) and the vector P* 1 obtained by parallel displacement
According to (IX.
80') the quantities
l
l
l
through the distance dx = U dr of the vector P (r) connected with the
l
point x (r). Thus the equations (9) express the fact that the four-
momentum
vector at the time t-\-dr
time T by a parallel displacement.
equations (9) may also be written
110.
Momentum and mass
is
obtained from the vector at the
On
account of (IX.
51),
the last
of a particle. Gravitational force
The equations
(9) or (11) determine the motion of a material particle
in a given external gravitational field. Strictly speaking, the particle
itself will create a gravitational field which should also be described by
In the present sections we assume, however, that this
comparison with the external field so that its influence
the functions g lk
field is
weak
in
g lk may then be regarded as known
functions of the space-time coordinates (x l ).
In order to provide a better understanding of the physical meaning of
on g lk
be neglected.
may
The
the quantities occurring in (11) we try to write these equations in the
form of three-dimensional vector equations. Let us define a spatial
vector p lt p by the equations
l
If p t
=p
construct a
t (t)
is
= mu
= mu =
l
mu
regarded as a function of the time variable
we can
new space vector
the spatial co variant derivative of the space vector p L with respect to t
Here the y^ lK are the three-dimensional Christoffel symbols formed by
.
means of the
spatial metric tensor
y lK and the vector dc pjdt
,
is
thus the
three-dimensional analogue of the four-vector (106), (11).
Remembering that
-~
r,
y LK
g iK +r,r^
(14)
ON PHYSICAL PHENOMENA
110
X,
easily seen that the equations (11) for
,
in the form
it is
if'
where
is
1, 2,
291
may
be written
= * = mG"
(15)
a space vector depending on the dynamical gravitational
potentials (y t x) an(i their first derivatives. (See also Appendix 7.)
If p l
mu l is interpreted as the momentum of the particle, L must
be interpreted as the (covariant) gravitational force acting on the
,
defined by (12) thus appears as
particle. The proportionality factor
the inertial mass of the particle moving with the velocity u l in the
K = mG m
also
gravitational field with the potentials (y t x). Since
L
L
represents the gravitational mass of the particle. For a particle at rest,
the mass reduces to
o
,
which thus represents the rest mass of a particle in a gravitational field.
For small particle velocities, where we can neglect terms of the order
ufc the quantity O is equal to a given by (VIII. 95), but in general
L
the gravitational force
will be a complicated expression containing
the potentials, their first derivatives, the velocity of the particle, and
even its acceleration. There is one important case, however, where the
L
becomes extremely simple, viz. if the system of coexpression for
L
ordinates is time-orthogonal so that the vector potentials y
0. In
t
this case
it is
immediately seen that the
identical with (15) if
<?
l=
first
three equations (11) are
we put
K=
_JJ*
OX 1
mG=
-mgradx,
(17)
the gravitational force is connected with the scalar gravitational
potential in the same way as in Newton's theory. This holds for arbii.e.
and for all velocities (see also Appendix 7). Further,
the expression (12) for the 'relativistic' mass reduces to
trarily strong fields
if
o,
On
the other hand, for weak fields where the dynamical potentials
(y x) may be treated as small, we find, neglecting terms of second order
in these quantities as well as in u/c, that (11) is identical with (15) if
t
we put
INFLUENCE OF GRAVITATIONAL FIELDS
292
^=
where
is
X,
110
i"""^
the space tensor defined by (VIII. 110) for the case of weak fields.
last term in (18) is of the type of a Coriolis force. On the rotating
The
disk considered in
90,
we have
x
in the system of coordinates
(x, y, 2, ct)
corresponding to the form (VIII. 83) of the line element, and in the
approximation corresponding to
(18)
-(
c
(18')
02-0
c
Hence
2
c
000
and
for the gravitational force
we
get from (18)
A = (m^x+Zmuu 2 ma^yZmwu
r
1
,
0).
'
(18")
'
For small distances r from the centre and for non-relativistic
2
2
particle velocities where terms of the order u /c can be neglected, the
gravitational force thus reduces to the usual combination of the
centrifugal force
and the
Coriolis force,
Usually, the y t (and their space derivatives) are smaller than or of
the same order of magnitude as x/c 2 (and its derivatives). For small
values of ujc the last term in (18) will then be small compared with
the first term. Further, if the field is stationary or quasi- stationary,
t
we may also neglect the second term
in (18), and the gravitational force
the
Newtonian
again given by
expression (17). Thus we see that, in
the case of weak fields and small velocities, the contribution of the
is
dynamical vector potential to the gravitational force is generally less
important than that of the scalar potential. This is due to the large value
of the constant c. The vector potential has an appreciable influence on
the motion of a particle only for strongly fluctuating gravitational fields.
If also the geometrical influence of the gravitational field is weak, so
that the physical space is approximately Euclidean, the motion of the
ON PHYSICAL PHENOMENA
110
X,
particle is the same as that of a particle acted
(17) in a system of inertia.
The
293
upon by a
force of the type
spatial co variant derivative of the contravariant
with respect to
is
defined
components p
by
Since the spatial metric tensor y lK may depend on /, dc p jdt and dc pjdt
will not in general be the components of the same space vector. A calcul
lation similar to that leading to (IX. 80'") gives here
or the reciprocal relation
*'*>'
yiA^/^v = &
dt
dt
K =
where
*Y**
pA
dt
(19 >)'
ft
KK
are the contravariant components of the gravitational force. The time
norm of the momentum vector is, by (19) and (19'),
derivative of the
dp,
dp
V+^dt
=d
c p,
dr p
dT p
nT r
- 8y
ll
(20)
In a system where the dynamical potentials are zero, y t
0,
%
the gravitational force vanishes, and the motion of the particle is given
by
(20/)
'
df*
i.e.
the covariant components of the
momentum
vectors at different
times are obtained by parallel displacements in the three-dimensional
sense. In general this does not mean, however, that the magnitude of
the
momentum
vector
is
we
constant in tim, for from (20)
p is constant only if our system of reference is rigid, i.e. if yIK
time-independent. If the dynamical potentials are zero, we have now
Thus,
is
get
~
m-
^
'
v2
mW -~
*S^
'
INFLUENCE OF GRAVITATIONAL FIELDS
294
and,
further the frame of reference
if
and the equations of motion
.e.
rigid,
(20')
may
=, 0.
both u and
ra are
110
constant
be written
(20"')
a geodesic line in physical
the particle is moving with constant velocity in the 'straightest*
this case the orbit of the particle is
Hence, in
space,
is
X,
compatible with the geometry of the space. The motion of the
particle is thus completely analogous to the motion of a free particle bound
line
move on
a smooth curved two-dimensional surface in a system of
where
the only forces on the particle are the normal reactions
inertia
of the surface. The only essential difference is that here we have to deal
a curved three-dimensional space.
with the motfon of a particle
If the spatial metric tensor varies with /, the motion of the particle
to
in the gravitational field is analogous to the motion of a particle on a
smooth variable surface in a system of inertia. Thus, if the dynamical
potentials are zero, the action of the gravitational field has the character
of a 'normal reaction' from the curved three-dimensional space.
111. Total
energy of a particle in a stationary gravitational
field
While the first three equations of the set (11) represent the equations
4
of motion of the particle, the fourth equation corresponding to i
must be the law of conservation of energy. In this section, we consider
only the case of stationary fields, where we have no flux of gravitational
energy, and leave the general case to a later section ( 1G). Hence the
g lk are time-independent
and the equation
^=
i
e.
I\
is
in question
becomes
(21)
0,
a constant of the motion in this case. The constant
which on account of
(4) is
cP^
of the form
may be interpreted as the total energy of the particle in the gravitational
field (see
Appendix
7).
For small
velocities
we
get, neglecting
terms of
second and higher order in u/c,
(23)
ON PHYSICAL PHENOMENA
111
X,
295
since the terms of first order in the velocities cancel. // represents the
energy of a particle in the field If we retain the terms of second order
rest
in u/c,
we
get, using (16),
H - # +|mX,
and
for
weak
(24)
reduces to the usual expression for the energy
with the gravitational potential x> i.e.
fields this
of a particle in a field
ff
AoC
+jA
tt
+AoX-
(25)
The last two terms represent the kinetic energy and the potential energy,
respectively. For large velocities such a decomposition of the energy
into a kinetic part and a potential part is not possible.
112.
General point mechanics
If the particle
acted upon by non-gravitational forces also, for
we have, on account of (IV. 55, 57),
is
instance electromagnetic forces,
a general system of coordinates
^^=0,
where
is
(27)
a four-vector which in a local system of inertia is identical
54). In the first instance, the physical
with Minkowski's four-force (IV.
meaning of the quantities
may thus be obtained by a transformation
of the physically well-defined Minkowski four-force. The components
may, however, also be given a simple physical interpretation in the
l
system of coordinates
equations (27) and
itself.
If
we put
F =
L
r$5n
J\=rg, -(3.u)).
dr.T,
we
get from the
(2)
i.e.
Since dt
(x
we
three equations (26)
find, in
may
the same
way
as in
(28)
110, that the first
be written
j
(29)
which shows that the space vector g with the covariant components
= FJT must be interpreted as the non-gravitational force on the
3f
t
particle.
INFLUENCE OF GRAVITATIONAL FIELDS
296
112
X,
Further, in the case of a stationary gravitational field, the fourth equation (26) can be written
tf *
~p
**>
dr
or,
by means of
(14), (22),
and
^jf
(28),
= (.u)=8fe*
(30)
Since the right-hand side is equal to the work done by the force g per
unit time, this equation expresses the law of conservation of energy.
Time -orthogonal systems of coordinates. Elimination of
the dynamical potentials
As shown in 94, it is not in general possible by a transformation of the
113.
type (VIII. 59) (i.e. without a change of the system of reference), to
ensure that the transformed vector potentials y\ or g^ vanish. The condition for this to be possible
that
is
"oc
where
W LK
0,
the space tensor defined by (VIII. 110).
However, if we allow arbitrary changes of the system of reference,
i.e. by means of a general space- time transformation
is
*'*
we can always obtain
=/*(**),
(31)
(32)
g'^
even for the most general type of gravitational
fields
described
by
1
lk
arbitrary functions g lk or g in the system (x ).
From
(IX. 7)
we
get, if (32) is satisfied,
'
n 'il n
9 941
t4
i.e.
0'
n'i*n
#44
(J
44
-0,
<7'
&l
d
4>
--!-
(33)
(/44
The transformation law of a tensor
^iH
J
in
connexion with the condition
t4
thus leads to the equations
</'
For f*(xk
we can now choose an
(34)'
v
'
'
2 3)
'
(35)
arbitrary function satisfying the
necessary condition (VIII. 50):
0'
= J- =
|, fl<7""
grad,/* grad'/
<
0.
(36)
ON PHYSICAL PHENOMENA
113
X,
For the functions f
equations of
first
we then
(x)
get the three linear partial differential
order
grad'/g
Each
of the functions/ thus
of the type
= 0.
(37)
satisfy a partial differential equation
%=
known
be considered
must
A =
where the quantities
may
297
(38)
>
grad*/
functions.
The general
solution of an equation of this type contains an arbitrary
function of three variables which may be three of the independent
variables or arbitrary combinations of them. For the functions /*,
we can thus take three arbitrary independent solutions of (38), and the
transformations (31) will then lead to a system of coordinates in which
1
g'
== 0'
i4
o or y\
0.
In this way the vector potential has been
'transformed away' and we have obtained a time-orthogonal system of
coordinates. On account of the simplifications which arise from the
disappearance of the vector potentials, in the future we often take advantage of this possibility and use time-orthogonal systems of coordinates.
In the preceding discussion, the function / 4 (x) was chosen arbitrarily
apart from the very mild restrictive condition (36). We shall now see
also possible, by a suitable choice of the function / 4 (a;), to
the scalar potential vanish. On account of (34), the condition x
that
it is
i.e. 0'
44
44
1/0'
4
,
i.e.
0>
leads to the condition
for the function
make
the
norm
(39)
of the gradient of the function f*(x)
1. Geometrically this means that the
must be constant and equal to
constant are at a constant
manifold of hypersurfaces defined by / 4 (#)
distance from each other. This can obviously be obtained in an infinite
number of ways, since we can choose one space-like surface arbitrarily
and construct all the consecutive surfaces such that the distance between
two surfaces measured along the normal is constant for all points on the
surface. Now, substituting the function / 4 obtained in this way in (37),
and solving these equations, we get a set of transformation functions
f
(x)
which leads to a system of coordinates in which the four dynamical
298
INFLUENCE OF GRAVITATIONAL FIELDS
X,
113
gravitational potentials are zero. In a system of this type, the gravitational force
occurring in (15) is zero, and the motion of a free particle
is of the type discussed on p. 293. If the initial velocity of the particle
is zero, we see at once from (20") that the particle will remain at rest in
this system. The different points of reference in the system may thus be
represented by an assembly of freely falling material particles, a circumstance which may be used in the practical determination of the trans-
formation functions/ (x).
Although a permanent gravitational field cannot be completely
transformed away except in infinitesimal regions of space-time, it is
1
thus always possible to eliminate the dynamical effects of a gravitational
field over finite regions of 4-space. The effect of the gravitational field
then purely geometrical and is completely described by the spatial
metric tensor. Although very interesting from the theoretical point of
view, the possibility of transforming away the dynamical properties of
is
usually of little practical importance, since the
of
reference is generally not rigid and the timecorresponding system
of
the
dependence
spatial metric tensor makes a treatment of physical
the gravitational
fields is
phenomena in this system very complicated. In the treatment of cosmological problems in Chapter XII we shall, however, make use of this
possibility.
(See
133, 134.)
Mechanics of continuous systems
By the same procedure as that applied in 109-12, all physical laws
of the special theory of relativity may now easily be written in a general
co variant form. In Chapter VI it was shown that the behaviour of a
114.
closed system in the special theory could be described by a symmetrical
energy-momentum tensor Tlk satisfying the equation (VI. 1) which in
the real representation
may
be written in the form
div,{7**}
ii
= 0.
(40)
The physical meaning of the
different components was explained in
Since the equation (40) is assumed to hold in any local system of
inertia, the general co variant form of the laws of conservation of energy
62.
and momentum must be
(see
IX. 92 c)
While the conservation laws of a closed system
in a
system of inertia
ON PHYSICAL PHENOMENA
114
X,
299
are expressed by an equation containing a sum of partial derivatives,
we see that this is not so in the general case on account of the term on
the right-hand side of (41). This indicates that the system is no longer
closed when placed in a gravitational field which itself may contribute
to the total energy
and momentum.
We
shall return to this question
126.
in
The components of the tensor T* in an arbitrary system of coordinates
S may be obtained from the tensor T* in a local system of inertia S by
means of the law of transformation of tensors and, since the physical meanthe same as in the special theory of relativity, we get, in this
way, a physical interpretation of the T% also. In S we can apply all the
considerations of
63, 64, i.e. if the physical system is so small that lk
ing of
5T
is
be regarded as constant over the whole, region occupied by the
system, we can unambiguously define the proper centre of mass of the
may
system. Relative to
motion relative to
S this point is moving with constant velocity and its
S will thus be like the motion of a freely falling
particle f
The three first equations (41) represent the law of conservation of
moment urn and the fourth equation is the law of conservation of energy.
If the gravitational field
(41) with
is
zero.
is
stationary, the right-hand side of the equation
Hence
0.
Appendix
shows that the determinant y
|y t
(42)
J formed by the com,
ponents of the spatial metric tensor (VIII. 64), is connected with the
determinant g
0,
\g lh
by the equation g
g^.y and, since g
<
>
0,
we have
(43)
The equation
(42)
may
thus be written
Now, putting
Ti
t A. D. Fokker, Proc. Amsterdam, 23, No.
Henri Pomcare, t. XI, fasc. V, 251 (1950).
5,
(45)
729 (1921). C. Moller, Annales de Vlnstitut
300
INFLUENCE OF GRAVITATIONAL FIELDS
and remembering that Vy
may
(44)
X,
114
time-independent in the case considered,
is
be written
-7
divS +
^=
(46)
0,
where
the three-dimensional divergence of the space vector S (cf. the analogous expression (IX. 90) for the four-dimensional divergence).
is
The equation (46) expresses the law of conservation of energy if h and S
are interpreted as the energy density and energy flux, respectively. In a
system of
local
As we
inertia, (45) is identical
126, the
shall see in
with the equations (VI.
momentum
density
is
2, 3).
similarly given
by
(47)
If the physical system
is
a perfect
the
fluid,
energy-momentum tensor
has the form
(48)
where
k
=^
(x
is
the four-velocity of the matter at the event point
and f) represent the invariant mass density and pressure,
(x ),
respectively, measured in a local rest system of inertia. The expression
(48) transforms like a mixed tensor and, since it reduces to the expression
(VI. 104) in a local system of inertia, it must be the correct expression for
while
j5,
yf in every system of coordinates.
For incoherent matter we have simply
V*.
(49)
By means of the general rule exemplified in
(IX. 84, 85), the conservation
TJ"
A(f
laws (41) can be written
(A
From
and
(6)
we get by covariant
since
we have
Thus,
if
^U-^+(A
^)-^./i
o-
(so)
differentiation
= U\ U
= 0.
l7.J7
U\ k
ltk ,
(51)
lffc
we multiply
by U we
l
(50)
obtain
o.
(52)
This equation, which is the generalization of the equation (IV. 211),
expresses the conservation of proper mass.
ON PHYSICAL PHENOMENA
114
X,
On
and
account of (IX.
79, 80")
301
we now have
thus reduces to
(50)
= 0,
/a.^
dr
(53)
is the generalization of (IV. 215) for the case in which the gravitational forces are the only forces acting on the matter.
comparison
with (IX. 80 1V ) shows that each infinitesimal part of incoherent con-
which
tinuously distributed matter
From
*
(45), (49), (2),
and
is
(4)
moving
we
like
a freely falling particle.
get for the energy density
=(54)
given by (3). Now consider a small piece of matter with the
Q
d l d& 2d8? in a local rest system of inertia $. On account
volume dV
where F
is
of the invariance of the four-dimensional volume element (IX. 47)
o
have, since the proper time r
is
identical with the time in
dx dx*dx*dt
S/(i</l)
By means of (43) and
we thus
(14)
we
$,
d& d$?dr.
get for the volume
dV of the material
particle
7T/
ft i/
Ct/
.j
<
\Jt\t
//'>**/7
x**v/ /y*"
irT/ Cl/t*/
Cc/vt/ Ct/C/
<
dF
dV9
"IU
fT2J
dV
dr
J*
7~f ~7zl
(55)
the generalization of the Lorentz contraction formulae in the
K
0, we get
presence of gravitational fields. For U
This
is
dV
dV,
(56)
the volume of a small piece of matter at rest in 8 is equal to its volume
o
in accordance with tha general assumption made on
measured in
i.e.
p. 223.
If T&O
o
^5<>
dV Q denotes the proper mass of the particle, its total energy
H = h dV is obtained from
(54)
and
(55)
(57)
in accordance with (22).
INFLUENCE OF GRAVITATIONAL FIELDS
302
The electromagnetic field equations
For simplicity we confine ourselves to the case
X,
115
115.
of electrodynamics in
the vacuum, the generalization of the macroscopic theory in ponderable
matter running along the same lines. By means of the generally covariant
expressions for the divergence and curl of an antisymmetrical tensor
(IX. 93, 94), we can write Maxwell's equations (V. 9, 13, 16) in an arbitrary
system of coordinates
form
in the
0-^1*
+
,
ki
+
,
3Fh
=o
*Q
'
(59)
F
Here, F
current density, p
kl
lk
of inertia, and
is
is
the electromagnetic field tensor, s l is the fourthe charge density measured in a local rest system
the four- velocity of the electric charge. In every
local system of inertia S, the equations (58) and (59) reduce to the
equations (13), (16), or (1) in Chapter V.
On
f/* is
account of the antisymmetry of
Flk
we
get at once from (586)
which
is
'
(60)
the general form of the continuity equation of electric charge.
t
x*/c lies entirely inside a
Now consider a system which at the time
region of volume V in 3-space. Multiplying (60) by *J\g\
l
2
integrating over the space coordinates x x x* we get, since the
finite
summation over
three terms in the
with respect to x l x 2
,
rr
and
first
are partial differential coefficients
3
,
Vlff
s*
dx*dx*dx*
0.
(61)
Thus the quantity
h+
^s*J7 dx dx*dx*==
[(
dV
(62)
a constant in time, which must be interpreted as the total electric
charge of the system. Hence
is
(63)
ON PHYSICAL PHENOMENA
115
X,
303
must be the charge density in the system $. Now consider the charge
o
Q
p dV connected with a volume element dV\ if dV represents the corresponding volume in the local rest system, we get from (63) and (55)
p
dV
dV Q
(64)
the electric charge of the small piece of matter considered
dent of the system of coordinates.
i.e.
is
indepen-
The invariance of the total electric charge also follows directly from the
continuity equation (60). Consider two arbitrary systems of coordinates
S and S' and the integrals
e=
J\g\ s*
dx l dx*dz*
(65)
x 4 -a
e'
l
<J\g'\8'*dx' dx'*dx'*
(65')
over the two regions defined by
#4
On
respectively.
and
(65')
we
constant
#' 4
and
constant
6,
account of the time-independence of the integrals (65)
any change in the values of e and e', choose b
can, without
so that the regions #' 4
a do not overlap inside the tube
6 and x*
in 4-space in which the charge density is different from zero. Then we
can introduce a third system of coordinates /8" which coincides with S
inside the region # 4
4
l
J\g* s" dx" dx"*dx"*
1
and, since x"
#'
6, (66)
Vl/
x'*^x*^a
x'*^jr' 4
x on the hypersurface x4
1
6.
As the
we have
equation (61) holds also in the system 8",
#' 4
a and with 8' inside the region
<*"
dx" l dx" zdx"*
(66)
=b
a and x'fi
x' 1 in
the region
leads to the equation
e
(67)
which expresses the invariance of the total charge.
On account of (59), (63), and (2) we have
H,
(68)
and the continuity equation
can be written
-=0
Vy
(60)
dt
or
divpu
H +
= 0,
^Vy
dt
(69)
INFLUENCE OF GRAVITATIONAL FIELDS
304
115
X,
which is the three-dimensional form of the continuity equation when the
spatial metric is time-dependent. For rigid frames of reference, where y
does not depend on t, (69) reduces to the usual form
In order to obtain a closer understanding of the physical meaning of
the tensor components F lk we shall now write the Maxwell equations
(58)
the form of three-dimensional vector equations. For simplicity
that our system of coordinates is time-orthogonal, i.e.
we assume
which, as was shown in 113, can always be obtained by a suitable choice
of the space-time coordinates.
Let us now introduce two antisymmetrical space tensors
B IK
and two space vectors
Fut
_ _#"_
"V(l+2x/c
and
..
'
__
'__
V(l+2x/
IK
and
by the equations
"
'
__&_
1+2X/C
'
(71)
These quantities are thus connected by the equations
H ___
_
IK
BIK
___
__
^V(l + 2 X /c
From
)'
^79^^
Vtl+Sx/c*)'
(IX. 27) and the corresponding equations for spatial tensors
then get, using (70) and
IK
T) __ __
(1
(j
vim
i/tjy/cw*
we
(71),
.,,
(J
php __ v
1/tXy KH L
ri
With these
notations, the equations (58 a), (586)
the form of three-dimensional tensor equations:
U-
may
be written
o,
r.
fir.
(74)
vy
bt
ON PHYSICAL PHENOMENA
115
X,
where
curl
curiu
^
curl...E
,
*A
IK
305
At
&EI
= #* -iWAC
-
ftr 1
(75)
divD
are the three-dimensional covariant differential operators. Further,
LK
defining the axial vectors dual to the antisymmetrical tensors
,
LK
by
**//!
(see
Appendix
Vy//
23
-^*-^|
=
= Vy^
//2
Vy//
31
12
//3
,76,
Maxwell's equations (74) take the familiar form
5),
div
CV/y
~~
(77)
},
,
curl
HT
(VyD)
-
\y
where curl
curl l/c
E and curlH
pll
!
,.
div
C ct
D=
~.
are the vectors dual to the tensors curl tK
H, respectively. For a
rigid
system of reference, where y
E and
is
time-
independent, these vector equations are of the same form as Maxwell's
phenomenological equations in ponderable matter and, since (72) can
be written
D=
E,
B - M H,
(78a)
:,
(786)
we
see that the gravitational field, besides its influence on the spatial
geometry, acts like a medium with the dielectric and magnetic constants (78*6).
116. Electromagnetic force
The electromagnetic
is,
and energy -momentum tensor
four-force acting
on a particle with charge
according to (V. 85),
F,
*
c
F*W<.
For the components of this vector we
3595.80
get,
(79)
by means of
(2)
and
(73),
INFLUENCE OF GRAVITATIONAL FIELDS
306
X,
116
A comparison with (28) shows that the electromagnetic force on a charged
moving
particle
is
(80)
which, by means of the axial vector
defined
by
(76),
be written
may
(81)
where u X
(u
is
x B)
the vector product with the covariant components
t
Vy(tt
-w 3 ,B
2
,
u*B l -u l
*>
u l B 2 -u*JB l
(81')
Appendix 5).
For a system with continuously distributed charge the force on the
charge in a volume element dV is given by (81 with e = p dV. Therefore
the force density f must be
(see
(82)
which
formally identical with the expression for the ponderomotive
force (VII. 73) following from Minkowski's energy-momentum tensor in
is
ponderable matter.
For the four-force density we get from
/>
which, on account of (68) and
lf(E l
(V. 93)
V,
(73), leads to
+B
(83)
the expressions
*><*.")
(84)
\
V(T+
which represents the generalization of (IV. 214) in the presence of a
gravitational
field.
According to (V. 96) the equations of motion, in the general system S,
are
If we multiply this equation
charged matter, then, since
type of force, we get, using
by the proper volumeodV of a small piece of
the proper mass /x dV is conserved by this
(55),
(87)
ON PHYSICAL PHENOMENA
116
X,
where
dV Q U
/l
the
is
account of (85) and
four-momentum of the small
(14), these
and
The
may
equations
= K+f dV,
%?
at
similar to (29)
307
*Z
at
also
(f .
particle.
On
be written
(88)
u) dV,
(30).
equations are the equations of motion of the small piece of
matter under the influence of the gravitational force K and the electrofirst
magnetic force
dV, the last equation
is
the law of conservation of
energy.
From the validity of the equations (V.
we get m the general system S
105, 106) in a local
system of
inertia
fl
-di
5*
where
Sf
S\
=F
F*
-^\(Flm F
(90)
the electromagnetic energy-momentum tensor. The equation (89)
consequence of the field equations (58) together with (83),
is
Using
(71), (73),
and
(76) in (90)
we
is
get for the different components
of the tensor S*
(91 a)
where
ExH
and
DXB
are the vector products defined
by (8 1') or by
and (6) in Appendix 5.
From (45), (47), and (91) we get for the electromagnetic energy flux,
-energy density, and momentum density
the equations
(5)
= -(DxB).
These expressions are
c(ExH),
= KE.D+H.B),
(92)
(93)
in complete agreement with the formulae (VII.
following from Minkowski's energy-momentum tensor
(VII. 68) in ponderable matter, as distinct from Abraham's tensor which
gives g
(ExH)/c instead of (93).
70,
71,
72)
INFLUENCE OF GRAVITATIONAL FIELDS
308
1
17.
Propagation of light in a
static gravitational field.
117
X,
Fermat's
principle
and y lK and x are indepenIn a static gravitational field, where y
dent of the time variable the electromagnetic field equations (77) take
the form of Maxwell's phenornenological equations in a medium at rest
t
with
Since the spatial geometry may be regarded as Euclidean in a sufficiently
small part of space, it follows from these equations that a light wave with
sufficiently small wave-length, i.e. in the limit of geometrical optics, will
propagate with the velocity
w=
= -p-- =
/(l+-%\
in accordance with the equation (VIII. 70).
in a
in
static gravitational field is therefore
(94)
The
trajectory of a light ray
the same way as
determined
an inhomogeneous refractive body, i,e. by Fermat's principle (see
according to which the time which the light takes to travel between
10),
two points A and B in space is a miiumum for the actual path chosen
by the light ray. Mathematically this is expressed by the variational
1}
^
principle
*^-
8j
for all variations of the curve connecting
our case, (95)
may
==0
(95)
r)
A and
B. Since y lK
g iK in
be written
L(x x )dX
i
---
(96)
with
x (X)
Here, A is an arbitrary parameter in a parametric representation x
of the curve considered. According to the equations (21), (24) in 86, the
condition (96) is equivalent to the Euler equations
l
M'LJl = M
dX^ig^x*^)!
.A
Since the parameter A was completely arbitrary,
,
that
w VtoicA*^) =
w*
~j(
w=
constant
dx c
we may choose
(98
'
l
it
so
(99)
ON PHYSICAL PHENOMENA
117
X,
along the curve determined by
dc u
_d
7T " dX
L
((/LK
^v
(98).
These equations then reduce to
^^.- __L^f - __1^X
w*dx^
~~
~~
309
uPSx*
2 dx'
MOO}
(
determine the trajectory of the
4
1
the right-hand side
(
/w )grad x determines
the deviation of the light ray from the 'straightest' line. According to
4
is the gravitational force
(17) this term is equal to (l//r )G, where
which together with
The term
light ray.
(90) comj)letely
071
on a unit mass, thus the
gravitational force
The same
result
is
light ray
zero.
may
he a straightest line only
\vill
=
we get from
the
first
field
~~
!^\-o
;
which shows that
\r-.i
By
0.
equations with
along the time track.
equal to 1. Hence,
the
he obtained from the equations (VIII. 87) for
the last of
0, w ~= c^(
j/ 44 ),
the time track of a light ray/|* Since y
these equations may be* written
Further,
if
(101)
4 in the case of a static
'
dX)
constant
a proper choice of A this constant can be
w *.t-~
made
(102)
1,
and, on account of (101),
which shows that the parameter A defined by (102) is identical with the
parameter defined by (99). For i = 1, 2, 3 we then get from (VIII. 87)
-~
d\
LK
iK
=^
~2 3x L
----2 fa 1
which, on account of (102), is identical with (100). Format's principle
of least time thus holds in every static gravitational field.
t T. Lovi-Civita, fiend. Acca<l
16, 105 (1918), H. Weyl, Ann. d
Lmcei (.5), No 26 (1017),
Phys 54, 117 (1917).
p.
458; Nuorv Ctmenlo
(6),
XI
THE FUNDAMENTAL LAWS OF GRAVITATION
IN THE GENERAL THEORY OF RELATIVITY
118. The gravitational field equations
IN the preceding sections we considered the influence of a given gravitational field on physical phenomena. We now turn to the most important
problem in gravitational theory, which consists in finding the general
equations determining the gravitational field variables (y (K y x) r the
g lk from a given distribution of mass. After several attempts, this
problem was finally solved by Einsteinf in 1915. In Newton's theory of
gravitation, the corresponding problem may be stated in the form of
,
Poisson's equation
where
6-664
4 &
(1)
x 10~ cm. gm.-
sec.~
(2)
the gravitational constant. This equation enables us to calculate the
gravitational potential x when the mass density /i is given as a function
is
of the space coordinates.
On account of the equivalence of mass and energy, we must assume
that any energy distribution, thus for instance an electromagnetic
field, will create a gravitational field. Now, the energy density of any
physical system is given by the component
tensor Tlk of the system, while x
2 c2
T44 of the energy-momentum
(l9w)
1S
connected with the
of the metric tensor; thus, equation (1) expresses the fact
that a certain differential operator of second order, acting on </ 44 is
component
44
proportional to 44 Since the general field equations must be covariant,
and since different components ofTlk are mixed up by a transformation of
coordinates,
it is
natural to assume that the general
^ _
m
the form
T
K -*iki
ik
field
equations are of
,*
('*)
where K is a universal constant, and lk is a tensor of rank 2 depending
on the metric tensor g lk and its first and second derivatives only. Since
the equation (3) for weak fields must reduce to Poisson's equation (1),
lk must be linear in the second derivatives of g lk and the only possible
expression for
lk is then of the form
ik
where
ct
and
c2
z=
R +c
lk
are constants, while
R.g lk +c^g lk
(4)
R lk and R are the contracted forms
of the Riemann-Christoffel curvature tensor defined by (IX. Ill, 114).
t A. Einstein, Berl. Bcr. % pp. 778, 799, 844 (1915),
Ann.
d.
Fhys. 49, 769 (1916).
THE GENERAL THEORY OF RELATIVITY
118
XI,
311
On
account of the symmetry properties of the tensors occurring in
(3) represent ten differential equations for the ten functions g lk A simple consideration shows, however, that these ten equations
cannot be independent (Hilbert).*)* Consider, for instance, the special
(4)
the equations
.
case of
empty
Tlk
space, where
0.
lk
The equations
(3)
then reduce to
0.
(5)
If these equations were independent, the ten equations (5) would in a
definite system of coordinates (x ) allow us to determine uniquely the
l
functions g lk (xt) throughout the whole 4-space,
and dg lk \'dy} are given on a hypersurface
when the
values of g lk
^=
If
constant
a.
(6)
w e now introduce a new system of coordinates
r
x' 1
xl
x'*(x),
x' 1
by
(7)
x*(x')>
the transformed functions
satisfy a set of differential equations
M'lk
0,
(9)
where, on account of the covariance of the equations (5), M'lk are the
same functions of g'lk dg'lk /dx' d*g'lk /dx' dx' m as M^ k of g lk dg lk /dz?
m
Thus, by the same argument as before, the equations
d*g lk /dx dx
l
enable us to determine uniquely the functions
of g\ k and dg lk /dx' 1 on the hypersurface # 4
x*(x')
(9)
the transformations
#4
.= a,
(7)
such that
x'
from the values
and if we choose
g\ k (x')
a,
in the vicinity of the surface
but arbitrary elsewhere, we have
on this surface, and glk (x' ) must consequently be the same function of
the variables (x /l ) as g^x1 ) is of (x1 ). This is, however, in contradiction to
1
(8), which shows that the dependence of g'rk on (x' ) for points sufficiently
4
far from the surface x
a will, in general, differ from the functional
1
relationship between g lk and
Thus, if the
the quantities
(x
).
equations are to be co variant, we must assume that
on the left-hand side of (5) and (3) satisfy four identi-
field
lk
This means that the solutions g lk of the field equations contain
four arbitrary functions corresponding to the four arbitrary functions in
ties.
D. Hilbert, GMt. Nachr.,
p.
395 (1915).
FUNDAMENTAL LAWS OF GRAVITATION IN
312
XI,
118
(7), which only change our space-time description,
but not the physical system which produces the gravitational field.
In fact, as seen in 113, it is always possible by a proper choice of the
the transformations
space-time coordinates to ensure that the four functions l4 have the
S l4 throughout space-time. The six independent
prescribed values
equations which remain after the introduction of the four identities
involving the quantities 3/?A are thus just sufficient to determine the six
remaining components g lK of the metric tensor.
.
As we have seen
momentum
thef rm
Thus,
if
114, the
in
theorems of conservation of energy and
of a material system in a general system of coordinates have
Th =
div (T{)
4
we assume that the four
the tensor
(10)
identities involving the
^ ^^
are
lk
-=0.
components of
the conservation theorem for a material system is a consequence of the
field equations (3) in the same way as the conservation of electric charge
(X. 60) follows from iMaxwelTs equations (X. 58).
We therefore assume that the differential operators
lk
on the
left-hand side of the field equations satisfy the four identities (11).
According to (IX. 86) and (IX.
and,
if
this
12)
to be identically zero,
is
we now
get from
(4)
we must have
q = -J.
Thus, putting
c2
=^-
A,
where A
lk
and the
field
equations
(3)
is
a universal constant,
K, k -\R<J>k ~\g tk
we
get
(4')
take the form
M,k-- K lk -\Eg lk -^ tk
- K T, k
-=
(12)
Writing down this tensor equation in mixed components, we get
3/{
contraction
By
we
where
is
obtain,
tf?--|#8{ -AS*
-*!*.
since R\ ~ R, S[ =
j?-f 4A - KT,
T=
(13)
4,
(14)
T\
(15)
the invariant obtained by contraction of the energy-momentum tensor
Eliminating R by means of (14) the field equations (12) can also be
Tlk
written in the form
.
(16)
XI,
THE GENERAL THEORY OF RELATIVITY
119
313
The linear approximation for weak fields
The field equations (12) and 1 6) are in general non-linear partial differential equations in the functions g lk However, for weak gravitational
119.
fields,
the field equations
may approximately be replaced by a set of linear
differential equations, f
weak
field
means that ue can introduce a
system of space-time coordinates in which the metric tensor
f/,A
is
of the form
,A+A,A.
(17)
where O lk is the constant metric tensor (VIII. 41) of the special theory,
and the h lk and their derivatives are small quantities whose squares
may be neglected. The ( hristoffel symbols (IX. 50) will then obviously
be small of the first order and, in the curvature tensor (IX. 99, 114), we
can neglect all terms depending on the squares of the Christoffel sym1
bols.
Hence we get
R '*
_^
~
~dx k
rs
=
^
/%< t ^A_J
p^+^^M--"
dx
dx \8x
rV
dx
dx
JL
2
.a*
2
ftE^af
xr dxs
i/ &*.
2\dx'dx
&%
a'AH
dx r dx k
dx dx r )'
where we have put
hi
Let us now
first
This means that
=G
rs
h ls
h rr
G'h rs
(19)
consider the case of a static distribution of matter.
the systems of space-time coordinates of the
in which the field variables li tk and the comwill
be
some
there
type (17)
of
material
the
energy -momentum tensor are independent of
ponents
among
For the component
t.
where x
jR 44
we then
get from (18)
the scalar gravitational potential defined by (VIII. 94) and
Laplace's operator. If, in the case of matter at rest, we neglect the
small contributions of the elastic stresses to the energy-momentum tensor
is
is
we
have, to a
first
approximation,
(21)
}
f A. Einstein, Berl. Ber., p.
688 (1916).
FUNDAMENTAL LAWS OF GRAVITATION IN
314
Thus we get from
(16)
with
= >.
Now we know that Poisson's equation
static
and
119
XI,
is
quasi-static gravitational fields
(22)
a good approximation for all
inside the solar system. We
can therefore conclude that the constant A must be so small that the
A-term
may
be neglected for
all
gravitational
phenomena
inside the
planetary system of the sun. The A-term can be of importance only
problems and in all other cases we shall therefore put
for cosmological
A equal to zero.
The
field
equations (12) then reduce to
R lk -\Rg, k
^-*T
(23)
lk .
Further, comparison of (22) and (1) shows that the constant K
be connected with the gravitational constant k by the equation
*c 2
1-87
X 10- 27 cm gm.- 1
must
(24)
Returning now to the case of a general weak field, we first remark
that the system of coordinates corresponding to the form (17) with h lk
small to the first order is still to a large extent arbitrary. Any trans-
^ ^ X ^8x
formation of the type
where Sx (#0
J
is
a function which
small to the
is
(25)
first
order, will lead to
an
expression for the transformed metric tensor of the same form as (17).
Since the left-hand side of (23) is small to the first order, the same must
KTlk which means that, to the approximation considered,
be treated as invariant under the transformations (25). Hence
be the case for
Tlk may
the components of Tlk may be put equal to the corresponding expressions
in the local system of inertia.
The expression
with
x?
The quantities
(18) for
R lk may now be written in
A*-i8*A,
Xi*
*i*~ !#,**,
the form
= *!
(27)
Xi k X? transform like the components of a tensor by linear
transformations of the space-time coordinates. On the other hand, it
is
always possiblef by a transformation of the type
(25) to ensure that
the x* satisfy the equations
I?
t See
D. Hilbert,
Gfitt.
(28)
Nachr., p. 53 (1917), and the solution (31) which satisfies (28).
XI,
119
For
Rlk
THE GENERAL THEORY OF RELATIVITY
R we
and
then get
*tfc
and the
315
/Z=*J = inA,
4DAifc,
(29)
equations (23) reduce to
field
These equations are of the same form as the equations (V. 26) for the
electromagnetic potentials, and the solutions of (30) which vanish at
'
X,i(*S
where
/(
**
'
analogous to the retarded potentials (V. 46)
infinity are therefore
x' l )A.
(x
~^^
C)
l~
<te'WW,
(31)
If we can prove that the functions defined
'
i=lT2,3
by (3 1 ) also satisfy the conditions (28), they must be the required solutions
of the approximate field equations (23).
The proof that the
solutions
(31) really satisfy theequations(28)runsexactly as the proof of the validity
of the Lorentz condition for the retarded electromagnetic potentials.
In the latter case, this proof is based on the law of conservation of electric
charge (see V. 41 ). In the present case, the validity of (28) follows in the
same way from the law of conservation of energy
to the approximation considered,
3T
may
k
a?
From
h lk
(31)
First
X\
hence,
we can now also
we get from (27)
= h\~2h =
-h,
by means of
(31),
arid
momentum which,
be written
<
32)
find explicit expressions for the, quantities
h lk
x^k+\G^k h
xifc-l^xj;
(33)
T'dV
*"-/
and
h lk
tk
g?
-?*
'*
dP,
(34)
ZTT J
where the prime in T'lk and T means that these quantities have to be
taken at the place (x' ) of the volume element dV = dx' l dx' 2dx' 3 and at
the retarded time tr/c.
r
120*
Simple applications of the linear equations for weak
relativity of centrifugal forces and Coriolis forces
fields.
The
To begin with let
mass density /a
us consider a static distribution of matter, where the
/*(#, y, z)
is
a given function of the space coordinates
FUNDAMENTAL LAWS OF GRAVITATION
316
x'
(x,y,z).
In this case we have, according to (21) and
^=
7
"44
8,48^,
-A
T=-A
2
.
IN
XI,
120
(34),
*c 2 C nO(x' y' z')dx'dy'dz'
4-
[X--X'|
(35)
KC 2 C
h
and
p<>(x
i/'
z')dx'd!/'dz'^
for the gravitational potential
we
get,
_,
on account of
(24),
the usual
expression of the Newtonian theory
(36)
Hence the
line
ds 2
element
is
of the form
k
(Glk +h lk ) dx*dx
(37)
where x
The
*s
given by
(36).
spatial line element
da^
is
/I--?
(38)
hence the geometry is only approximately Euclidean and the coordinates
x, y, z are not exactly Cartesian. In general it is not possible by a change
of the space coordinates to introduce Cartesian coordinates. However,
the deviations from Euclidean geometry are in most cases too small to be
measured. At the surface of the earth, for instance, the quantity 2^/c 2
of the order of magnitude 10~ 9
For a system of material particles with the masses M\, M%,... situated
at the places X^
2 ,..., respectively, we get from (36)
is
(39)
For a
single particle the potential
symmetrical.
and the
line
element are spherically
We have
+^
(2k
1
r=|x-X|
(40)
XI,
THE GENERAL THEORY OF RELATIVITY
120
317
In the same way, the case of a stationary mass current distribution
may be treated by means of Einstein's approximate equations (34).
-
Thirring and Lensef calculated, for instance, the influence of the rotation
of a central astronomical body on the gravitational field and the corre-
sponding effects on the motion of the satellites. All such effects are too
small to be observed J and we shall not consider them here.
There
is
one effect of this kind, however, which, although small, is of
it throws new light on the nature and
theoretical importance since
origin of the centrifugal and Coriolis forces appearing in a rotating
system of coordinates N. According to the idea of Einstein underlying
the general principle of relativity (cf.
82), these forces are gravita-
tional
forces originating
from the rotation of the distant
celestial
masses relative to $, and such 'non-permanent gravitational
should satisfy the same general field equations as the permanent
The approximate
solutions (34) for
fields
fields.
weak fields do not directly allow us to
we may expect that
treat the effects of the distant celestial masses, but
a rotating spherical shell of uniform mass density will produce effects
inside the shell similar to the rotation of the distant celestial masses.
If the shell is at rest, the potential x given by (36)
the shell and has the value x
kM^R, i.e.
is
constant inside
>v
M
= - K *%,
*-r 2
4?r
(41)
is the total mass and R the radius of the shell.
2
2
which can be refrom
the constants 1
Apart
2^/c and 1 + 2^/c
and
time coordinates, the
moved by a simple change of scale of the space
line element (37) thus has the same form inside the shell as in the special
theory of relativity. In the case of a shell moving with constant velocity,
which can be reduced to the former case by a Lorentz transformation,
the line element inside the shell must therefore again be of the special
relativity type. This is also easily seen directly from (34), using the
expression for Tik in the case of stationary rectilinear motion of the
matter. This illustrates the fact that the uniform rectilinear motion of
where
the distant celestial masses relative to the different systems of inertia
does not give rise to any gravitational forces in these systems.
For a rotating shell of matter, however, Thirring found the interesting
by the
result that the field in the interior of the shell, as determined
equations
(30), is similar to
t
j
the
field in
a rotating system of coordinates,
H. Thirring and J. Lense, PJnjs. ZS 19, 156 (1918).
Do Sitter, Monthly Notices, 76, 699 (1916), 77, 155
H. Thirring, Phy*. ZS. 19, 33 (1918); 22, 29 (1921).
(1916).
FUNDAMENTAL LAWS OF GRAVITATION IN
318
120
XI,
thus leading to gravitational forces similar to the usual centrifugal and
Coriolis forces. We shall here consider the somewhat simpler case of a
Q and radius R, which is rotating
rotating massive ring of rest mass
clockwise in the xy-pl&ne with angular velocity to. If (#',?/', 0) are the
we have
coordinates of a point on the ring,
42 >
/7T44
-*
From
(IV. 198)
we
Hence, by means of
get for the total rest mass
is
2)
C f^-*,
J
P
- J{( x - x + y - y >)*+ z
')*
the distance between the point
ring.
of the ring
(34),
A lfc (*,y,
where
(x, y, z)
Further,
(43)
(44)
*}
and the
line
element ds on the
and
2
c2
c2
o>
y'x'
2
,
1
-
c
1
JP2
i-i*-?..
o
i+i
2
2
'
(45')
We
shall confine ourselves to the consideration of points (x, y, z) whose
distance from the origin is small compared with R. This means that we
can use an expansion of !//> in terms of x/R, y/R, z/R. Neglecting all
terms of higher than second order in these quantities and putting
COS,
XI,
we
THE GENERAL THEORY OF RELATIVITY
120
get
1 /,
319
# 2 +v 2 +z 2
a
,
00
x
V mi?
^ +r
+2 ^
a/
<w
'+-
(46)
Using this together with ds = R d&' in (43) the integration over
from to 27T can be easily performed. We shall- only write down the
<7l4 -f/i l4 which determine
expressions obtained for the components g i4
#'
the dynamical potentials,
i.e.
CTTOJ
^34==
(47)
where
3f
==
3f (l
RW/c
)-*
the relativistic mass of the
shell. These expressions should hold for all
which Rwjc < 1, the only approximations introduced
being those arising from the use of the equations for weak fields and the
is
values of
for
expansion of 1/p in terms of x/R, yjR, zjR.
For a ring of mass
at rest we get in the same
In this
case, the scalar potential is
way
thus not constant as inside a closed
shell. Hence, the contribution to the dynamical potentials arising from
the rotation of the ring is, apart from the unimportant constant term
E
in the expression for
a>
gr 44 ,
MKC*
a)
<-->
,
AN
FUNDAMENTAL LAWS OF GRAVITATION IN
320
comparison with (X.
moving
is
plane
XI,
120
shows that the gravitational force on a
heavy wheel rotating clockwise in the in/-
18')
test particle inside a
of the same type as in a system of coordinates
rotating counterIt is true that on account of the smallness of the
quantity
clockwise.
MKC 2 /^7rfi
the effect is too small to be measured, which explains the
negative result of the experiment actually performed by Friedlander ,f
b'ut anyhow the above considerations
suggest a connexion between the
in the world, and the
gravitational constant /c, the total mass
distance R of the distant celestial masses, of the type
MKC*
mean
It is interesting that the
dependence on the angular velocity of the
gravitational forces inside a rotating wheel is exactly the same as in a
rotating system of reference. The vector potentials y t in (48), which give
the Coriohs forces, are even of the usual form as regards their
dependence on the coordinates (x,y,z). On the other hand, the scalar
rise to
potential in (48) contains besides the usual term
on
x*+y 2 a term depending
This term gives rise to an axial component of the 'centrifugal'
which tends to drag a test particle into the plane of the rotating
wheel. One could think that this unexpected deviation from the usual
centrifugal force is due to the particular mass distribution which we
have considered. However, the calculations of Thin-ing show that the
z*.
force
same
effect appears inside a
homogeneous rotating
shell.
The purely
radial character of the usual centrifugal force therefore rather indicates
that the approximate
equations (30), which require explicit assumpboundary conditions at infinity in order to give
do not give an adequate description of the world as a
field
tions regarding the
unique solutions,
whole.
Firstly, the exact equations (12) are non-linear and, secondly,
contain
the A-term which for cosmologicacl distances may be of
they
Jn
importance.
fact, this term entirely changes the character of the
field
equations, in particular as regards the question of the boundary
conditions (see
132). But even if we confine ourselves to the linear
approximation of weak fields, a simple calculation shows that the
use of the A-term introduces into the solutions of the
equations for
the case of a rotating wheel extra terms which are of the same
type
as those occurring in (48), but
multiplied
on
2
(jJ
+j/
2
)
and
22
by
\R 2 The terms depending
do not, however, have the same
ratio as in (48); there-
l,
t B. and T. Fnedldnder, Absolute und relative Bewegung, Berlin, 1896.
(50)
THE GENERAL THEORY OF RELATIVITY
XI,
120
it is
understandable that the terms in x which contain z 2 may cancel
left with a purely radial centrifugal force. This con-
321
and that we are
sideration is, of course, very rough, since the linear equations probably
represent a bad approximation when applied to the world as a whole;
but, as we shall see in
133, the relations (49) and (50) also follow from
Einstein's solution of the exact gravitational equations applied to an
model of the universe.
ideal
by means
we
treat the case of a non- stationary distribution of matter
of (30) and (34), the close analogy to the wave equations of
Finally, if
electrodynamics leads at once to the result that fluctuating matter in
general gives rise to the emission of gravitational waves travelling
with the velocity of light and carrying with them a certain amount of
As shown by
energy.
emitted in this
way
Einstein, f the gravitational energy (see
126)
is, however, too small to give any measurable
astronomical effect.
121. Equivalent
systems
of coordinates.
Systems with spherical
symmetry
Let
g lk
be an arbitrary system of coordinates with the metric tensor
1
g lk (xl). If we introduce a new system of coordinates x' by
l
(x
x' 1
x' l (x*),
(51)
the transformed components of the metric tensor
(x>r)
f\^l
wm
W* w***W
<
are not in general form-invariant functions of the coordinates,
g\ k is generally not the same function of the transformed variables
as are the g lk of the variables (x ).
Two systems of coordinates (x 1 )
52)
i.e.
1
(x'
and
(x'
for
which the components
of the metric tensor are form-invariant functions of the space-time
coordinates under the transformation (51) may be called equivalent, since
any physical process will have the same course of development in the
two systems. The existence of equivalent systems of coordinates imposes
a certain condition on the gravitational field, since the functions (7i&(#0
obviously must satisfy the functional equations
In some cases the gravitational field variables glk are form-invariant
under a whole group of transformations. This is, for instance, the case
t A.
3595.60
Emstem,
Berl.
Ber
p. 154 (1918).
FUNDAMENTAL LAWS OF GRAVITATION
322
IN
121
XI,
non-permanent gravitational fields, for, if we first introduce
pseudo- Cartesian coordinates X by the transformations
for all
X*=/^),
we have
ds*
- glk dx dx =
k
(54)
Glk dX*dX
(55)
Further, by performing a Lorentz transformation
JT*
A>k
Xk
(56)
and afterwards introducing new space-time coordinates
the transformations
v//
.A
with the same functions/ as in
__
ft{
-G
k
g lk dx*dx
lk
and, since
g'lk
dX*dX k
(54),
by means of
/
r7
(*<
we have
G'lk
G lk
G[ k
,K
J (%
(x'
dX'*dX' k
g\ k
dx'*dx'*,
constant,
must be the same functions of the
as are the g lk of (rz ).
(a'*)
The
defined by (54), (56), and (57), which may be
called a generalized Lorentz transformation, thus connects two equivalent
systems of coordinates, and the components y lk of the metric tensor in a
transformation
x' 1
x' (x l )
l
non-permanent gravitational
field
are form-invariant under the group
of generalized Lorentz transformations.
In the case of permanent gravitational
fields,
generally not
that the g lk are
is
it
possible to introduce such space-time coordinates x
form-invariant under the group of four-dimensional orthogonal transformations, but in some important cases the gravitational potentials
l
are form -in variant under the sub-group of spatial orthogonal transformations. Such systems are naturally called spherically symmetric. Putting
xl
(x,c)
(x,y,z,ct),
the g tk are then form-invariant under any
orthogonal transformation of the three variables x, ?/, z with constant t.
Generally the coordinates (x,y,z) will not be Cartesian, the spatial geo-
metry being in general non-Euclidean, nevertheless, the line element
ds 2 = g ik dx l dx k can in this case be a function only of the well-known
foi'm-invanants of the group of three-dimensional rotations in a Euclidean space, These invariants are
r
dx
+dy +dz
x dx-\-y
dy-\-z dz
Since the line element
is
x *+y*+ z *}\,
=.
dr*+r* d0
r dr,
dt,
+r
dr
2
sin
and
\
2
rff
1.
(58)
a quadratic form in the differentials, the most
THE GENERAL THEORY OF RELATIVITY
121
XI,
323
2
general possible expression for ds in a system with spherical
is therefore
ds 2
2
F(r,t) dr +G(r
2
t
t)(r
d0 2 +r 2 sin 2
+ 2H(r,
where F, G, H,
may
drdt+L(r,
t)
dt 2
(59)
only.
t,
be further reduced by a suitable choice of co-
ordinates. Introducing a
tion
we
and
are functions of r
This expression
t)
2
d</> )
symmetry
new
variable
r' 2
r*G(r
see that the line element in the
instead of
r'
by the transforma(60)
t),
new
variables
ds 2 -- M(r',t) dr f2 +r' 2 (d0 2 +sm 2 0d<f> 2 )
is
of the form
+ 2N(r',t) dr'dlt+0(r',t) dt
2
.
(61)
The
present case.
Hence
ordinate clocks to
(VIII. 110) are obviously zero in the
possible by a simple change of rate of the co-
defined
o> tK
quantities
it is
make
by
the vector potentials disappear, and since
and <, this may be obtained by a time
N(t',t) does not depend on
transformation of the form
(62)
t'=f(r',t),
which will not affect the second term in (61). Dropping the primes, the
line element may thus be expressed in the standard form
ds 2 == a dr 2
where
are functions of r
a(r,
+r
t),
(d0
b
+sm
b(r,
bc 2 dt 2 ,
2
d<f>
t)
(63)
= +C"
I
and t which, on account of (VIII.
and t.
(64)
52),
must be
positive
for all values of r
122. Static
systems with spherical symmetry
If the system is static and spherically symmetrical, the functions a
and b are independent of /, and in this case the components of the tensor
lk
defined
Vu
by
(4')
aM>
are easily calculated.
=
022
r *>
I/as
With x
2
(r,6,(f>,ct)
r sin 0,
44
==
-6,
we have
(65)
all other components vanishing. The coordinate system is orthogonal;
the non-vanishing components of g lk are therefore simply
(66)
FUNDAMENTAL LAWS OF GRAVITATION
324
and the
Christoffel
IN
XI,
12J
symbols (IX. 50) are of the form
(no summation over i and fc ').
Since the g lk are independent of
and a: 4 we see that the only independent non-vanishing components of T lkl
T\ k are the following:
pi
In
a
_
~'
r? 2
1
r 22
-,
~'r
1
r
I
3s
-sin
</>
---,
cos e,
<5in2/?
r? 3
]
,
1
r
I
_
-
r|3 -=
r? 4
cotff,
=
lo
(68)
where the accents denote differentiation with respect to r.
Using (68) in (IX. 114) we obtain the tensor R lk from which we get
R\ by contraction. Finally, the tensor
lk and the
A
mixed
M*
are
from
obtained
(4').
corresponding
components
straightforward consideration shows that the non-diagonal elements of
this tensor are zero and the four diagonal elements M\, M\, M\, M\ are
functions of r only. From the transformation properties of tensor comand M\
\
ponents under rotations it then follows that the components
must be equal. However, on account of the identities (11), even the
three quantities M\, M\ = M\ M\ cannot be independent. As is easily
the scalar
seen, the equations (11) reduce for a static spherical
system to essentially
one equation, only, by means of which the components
3 may
\
be expressed in terms of M\, M\, and dM\/dr. The calculations give the
M =M
following expressions for the non-vanishing components of
M*.
"""'
Ml- Ml- -2
2a[\b)
-
a b
+
2\6 /
+ br
These explicit expressions for the components of the tensor
seen to be in accordance with the identities (11).
-A.
are easily
THE GENERAL THEORY OF RELATIVITY
123
XI,
325
123. Schwarzschild's exterior solution
In the empty space surrounding a material particle of mass
and the field equations (13) reduce to
have T* =
M we
r-
r2
abr
v
(706)
;
a)
1
[76Y
la'
1/6V
6'
^>
the last equation being actually a consequence of the two
From (70 a) and (706) we get by subtraction
a'b+ab'
of~~~
a z br
i.e.
(ab)'
ab
or
_
~
(71)
(72)
(73)
r2,
by
ry'+2/-l + Ara
By
integration
Ar 2
(yr)'
we thus
equations.
constant.
(706) becomes, after multiplication
or
C)
Putting
first
(7
(74)
(75)
get
A? 3
'iy
where
is
(73)
-- 2m,
(76)
a constant of integration, or
From
and
(77)
we
=!_*?*!.
y
o
(77)
get the following solution of (706).
(78)
The
of (63)
spatial line element
is
we
which
in
our case
is
equal to the spatial part
thus
2
neglect the small A-term, da will in the limit of large r go over
the
usual
line
element
into
for a Euclidean space in polar coordinates,
If
FUNDAMENTAL LAWS OF GRAVITATION
326
IN
XI,
123
and it is interesting that the condition of spherical symmetry is sufficient
to secure this result without any explicit use of boundary conditions at
infinity. This result is, of course, also partly connected with the normalization (60) of the variable r which has been chosen so that the geometry
constant is the same as on a sphere of radius r in a
on a surface r
Euclidean space. In the actual space (79), r will, however, not simply
be the radial distance, the distance between two points (r^d^) and
(r 2 0,
,
(f>)
measured with standard measuring-rods being
I
An
try to
(1
Ar 2 /3)~* dr.
2m/r
(80)
observer at great distance from the central body will, however,
make a picture of the system in a Euclidean space in this
;
picture the quantity r plays the role of the distance from the centre, and
the distance / measured by standard measuring-rods has 110 real impor-
tance in astronomy. The coordinates (r, 0, (/>) may therefore be taken to
be the usual polar coordinates applied in celestial mechanics.
According to
,
\)
(72)
we have
constant
=_
-^
constant X
/.
2m
Ar 2 \
/01X
(81)
3;
By a simple change of the time-scale the constant may be made equal
to 1, and we thus arrive at Schwarzschild's exterior solution |
/
^ /2
2m/r
.2/,/#2
2fl
/]
7J,2\
Ar 2 /3
^W
(82)
or, if
we
values of
neglect the A- term which
is
of importance only for very large
r,
2m IT
dt*.
(83)
This expression must hold outside a spherical distribution of matter
r determined by the equation
and, since it is singular at a distance r
1-^ =
0,
(84)
ro
we may conclude that the 'radius' of the mass must be larger than the
value determined by (84). The singularity at r = r cannot be completely
removed by the use of other 'static' coordinates. It can, however, be
f
K. Schwarzschild,
Berl.
Ber
p. 189 (1916)
THE GENERAL THEORY OF RELATIVITY
123
XI,
modified, for instance, by using 'isotropic' coordinates
r', 6,
327
<,
defined
by the transformation
r
The
line
'
+ -p'
r>
\{(r*-2mr)*+r~m}.
element then takes the form
(85)
^
(86)
The singular point
2m
has the new radial coordinate
rQ
|m, and
seen that in (86) the singularity has been removed from the spatial
part of the line element, but it appears again in (/ 44 which vanishes at
the same place. As shown by Sermi, Einstein, and Pauli,f no nonit is
empty space exist which are
1
and
have
the
form
^constant/r at infinity. As we
g^
stationary
shall see presently, this form of </ 44 indicates that the field is produced
by a mass distribution around the origin. The scalar potential which, of
singular solutions of the field equations for
is invariant under the transformation (85) has the following forms
the two systems of coordinates corresponding to (83) and (86)
course,
in
"=
(-1/44-
me 2
c2
0^
--
(8?)
'
c2
we*
(-^4-1)3-
~/ (H
l /2 /)2J
At large distances where the field is weak, both expressions reduce to
me 2 //', which shows that the constant
the Newtonian form
mc'2 /r
m must be connected with the mass of the particle creating the field
by the
,/
relation
m=
i
L->
1/f
vl
^^
C"
In the regions where the
field is
<>
-*
ifC* VI
KCjXl
-.
(88)
877
weak, (86) reduces to the expression
where we have put
x
r'sin#cos0,
?/
r'sin#sin</,
/'cos^.
(86")
On
account of (88) this expression is identical with the static solution
the linear approximate field equations (30)
of
(40)
As first remarked by Lernaitre, J the singularity disappears in the line
1
Senni, Atti Accad Lmcei (5), 27 , 235 (1918), A Einstein, Revista (Unw Nac.
11
A
Einstein
and
2,
Tncuman), A,
Pauh, Ann. oj Math. 44, 131 (1943).
(1941),
A, 53, 51 (1933), see also J L.
J (i E Lemaitre, Ann *S oc. Scient. Bruxellex, S'*i
Syngo, Proc Roy Insh Soc 53, No 6, 83 (1950)
FUNDAMENTAL LAWS OF GRAVITATION
328
element
if
we introduce a
non-static
IN
system of coordinates
123
XI,
r', 0,
<f>,
t'
by
the transformations
r
The
line
dt'
(9m/2)i(r'-cO,
dt~ (2m/ r)
dr.
(89)
2m/ r
element (83) then takes the form
ds*
^dr' 2 +r 2 (d0 2 +sin 2
2
rf<
)-c
eft'
(90)
depends on r' and Z' by (89). The new system of coordinates is
of the type considered in
113, where the dynamical action of the
gravitational field has been transformed away, the dynamical potentials
(y x) being zero. The motion of a planet is described in these co-
where
by the equations (X. 20'), i e. the covariant components of
the momentum vector of the particle at the time t'+dt' are obtained from
ordinates
the corresponding vector at the time
space with the line element
da*
by
parallel displacement in the
9m
dr' 2
+r 2 (d0 a +sin 2 0d^ 2
).
On
account of the time-dependence of r the system of reference is not
rigid, however, and the motion of the planet in these coordinates is
therefore rather complicated.
124. Schwarzschild's solution for the interior of a perfect fluid
The energy-momentum tensor of a
perfect fluid
is
given by (X. 48)
(91)
On
account of the
field
equations
(13),
M\ =
and the expressions
(69) for the tensor
icTf,
(92)
Jf J, we see that a static spherically
of the type (63) is possible only if the velocity u of the
symmetric
matter is zero and if the proper mass density /t and the proper pressure $
l
field
are functions of r only.
Hence we have, according
to (X.
2, 3)
and
(65),
(92')
THE GENERAL THEORY OF RELATIVITY
124
XI,
Using
with the conservation law
this
1
*
mk
J. *
get for
k\
d(J\a\T
/
^VV I"
I
If
A
\J
T*r _ mis
J.
'
we
J[
-,
/QQ\
7O
I
f^fn
llftl
-ri,*
or,
329
by means of
(68)
and (IX.
69),
This equation gives the dependence of the pressure on the scalar gravitational potential in the equilibrium state of a fluid under the influence
of its own gravitational field. The other three equations of the conservation laws (93) do not give anything new.
equations (92) again reduce to only two independent equations for which we may take the equations
The
field
M\ - - K T\,
Mi - -KTl
(95)
the other equations (92) being consequences of (95) on account of the
conservation equation (94). From (69 a, 6), (92'), and (95) we thus get
=
~\ll --}+*
a
r2
abr
(96a)
<$,
The equations
(94), (96) together with the equation of state of the
matter which gives the connexion between p and
determine the
/St
and the gravitational field of the fluid.
For simplicity we shall assume that the fluid is practically incompressible. The proper mass density may then be treated as a constant
and the solution of (966) may be obtained from the solution (78) of
interior state
(706)
by the substitution
A->A+^c
2
.
Since our solution
regular as r->0, the constant of integration
equal to zero. Hence
we
in (78)
to be
must be put
get
1
where
2m
is
_ ^ Qr
JP
r2
^-,.
1
J?2
(98)
FUNDAMENTAL LAWS OF GRAVITATION
330
Further, on account of the constancy of /t, we get at once
of (94)
2
constant.
p)v 6
(/ic
By
IN
by
XI,
integration
addition of the equations (96 a, 966) and multiplication by V6,
therefore have
-7,
constant.
a*r
Substituting in this equation the expression (97) for a
we obtain
^WL^I^^A,
is
a constant. Putting y
instead of r, (99)
(99)
dr
where A
x
(1
we
/,
a\or
124
V6 and introducing a
may also be written
r 2 /# 2 )*
y-xfy =
new
A.
variable
(100)
(ji/JC
The
solution of (100)
where
is
Finally, using (97)
for the pressure p
ds*
=A
Bx,
a constant of integration.
Hence
Thus we
is
2
?/
--
(.4
B<J(\-
r 2 /fl 2 )) 2 .
(101
and (101) with (96 a), we get the following expression
measured in a local system of inertia
arrive at Schwarzschild's interior solution f
2
(103)
The
spatial
geometry
is
defined
by the
line
element
8
).
(104)
=^ constant is the same as on a
?
Hence, the geometry on the surface r
l
r l is not the distance to the
of
r
in
radius
Euclidean
but
sphere
l
space,
origin r
measured by standard
rods, this distance being
(105)
t
K. Schuarzschild, Bert Her
p 424(1916).
XI,
THE GENERAL THEORY OF RELATIVITY
124
331
t
The volume of this sphere
TT
r\
is
2rr
r with a
Consider a sphere of fluid filling the space inside r
r 1 we then have the solution
constant proper density of mass /2,. For r
(103), while Schwarzschild's exterior solution (82) must be valid for
<
>
and
rx
We now
have to adjust the constants
(82) coincide for r
r v further,
and
so that (103)
has to be zero at the surface of the
sphere. If we neglect the A-term, which anyhow has only a small effect
inside the solar system, these conditions lead to the equations
with the solutions
A =
(107)
comparison with (88) shows that the gravitational field of the
spherical fluid at great distances corresponds to a mass
M = ^ r?A
r
which
is
(108)
thus the same as for a constant distribution of Newtonian mass
over a sphere of radius
r l in
a Euclidean space.
According to (106) the real volume of the sphere
is
larger than ~-r\
on the other hand, /l was the mass density measured in a local system of
inertia which differs from the mass density in the system of coordinates
is exactly
used here. Actually we shall show in 128 that the quantity
equal to the total energy of the system divided
difference between the real
volume
Vl
by
c2
Anyhow, the
given by (106) and ^-rl
is
in all
astronomical applications very small. In the case of the sun, for instance,
we may put
/t
3
l-4gm. cm.-
rl
6-95x 10 10 cm.
(109)
FUNDAMENTAL LAWS OF GRAVITATION
332
Hence we get
(110)
Also the difference between the distance
rl
the exterior solution (83)
satisfied; for
amply
...
As
in the case of
/?1 r+-> ..
A
lfj~6
1U
<.
-?&2 r^
empty space we can
For arbitrary functions
obtained by a transformation r'
^the general solution of which
2m
rl
coordinates.
>
by (105) and the
by the astronomical
defined
determination of r v
We also see that the condition
2m
124
2X10- 3
radial coordinate r x is far too small to be detected
'
XI,
3-5X
is
IN
1
1
for the applicability of
from (107) we get
also here introduce isotropic
a(r), b(r) in
(63), this
may
be
= r (r) satisfying the differential equation
f
VWy,
(111)
is
(112)
where
C is an arbitrary
constant.
The
line
element then takes the form
(113)
=
where
For
L. (dx*+dy*+dz*) - 6c 2
(x,y,z) are given
a(r) of the
and the
line
form
by
(97)
dt*
(86").
we thus
get
element inside an incompressible
fluid takes the
form
The constant C can be determined so as to make (114) coincide with (85) at
the boundary of the
fluid.
THE GENERAL THEORY OF RELATIVITY
124
XI,
333
Schwarzschild's solution represents the only exact solution of the
gravitational field equations which has found any application in astronomy. Reissnerf and WeylJ also solved the problem of the* gravitational
produced by the electromagnetic energy in the surroundings of a
charged particle. The result obtained by these authors is a line element of
field
the form
l~2m/r+Ke
'
/r
(116)
The
two terms depending on the charge and
thus, on account of (88),
ratio of the
respectively,
is
K6 2
477-e
the. mass,
(111)
'
rMc 2
r.2m
In the case of an electron, the two terms will therefore become of the
same order of magnitude at a distance corresponding to the classical
2
2
Both terms will, however, have negligible
electron radius a
e /Jfc
on the interaction between electrons as compared with the
Coulomb interaction.
effects
Further, exact solutions of the field equations for the case of arbitrary
cylindrically symmetrical distributions of matter
and by
125.
were given by Weyl
Levi-Civita.||
The
variational principle for gravitational fields
be an arbitrary domain in 4-space. Consider the four-dimen-
Let
sional invariant integral
Jl==
RdZ= R
(
where
Rlk g lk is
k
lk g*
l
<J(g) dx dx*dx*dx\
(118)
the curvature scalar and
is the contracted curvature tensor. The integrand in ( 1 1 8) is an algebraic
function of the g ik and their derivatives. Since the contra variant comlk
ponents g are uniquely determined by the gik the integrand can also be
lk
expressed as a function of the g and their derivatives. Let us now con,
sider a variation of the metric tensor 8g lk
t
t
||
which
is
arbitrary inside S,
H. Reissner, Ann. d. Phys. 50, 106 (1916).
H. Weyl, ibid. 54, 117 (1917) Raum-Zeit-Materie, 3rd ed. p. 223,
H. Weyl, Ann. d. Phya. 54, 117 (1917); 59, 185 (1919).
;
T. Levi-Civita, Rend. Accod. Lincei
(5),
Berlin, 1920.
No. 26 (1917), No. 27 (1918); No. 28 (1919).
FUNDAMENTAL LAWS OF GRAVITATION
334
but which vanishes together with the variation of the
at the
first
boundary of S. The corresponding variation of Jx
S/!
By
IN
%Rik J(~g)g lk
variation of (119)
we
dx+f
is
XI,
125
derivatives
then
R lk S(V(-W) dx.
(120)
get
(121)
While the Christoffel symbols T\ transform according to the transformation equations (IX. 53) which, on account of the first term on the
right-hand side, differ from the transformation law of tensors, the variation SFJ.,, which is the difference between two Christoffel symbols at the
t
same
point, will obviously transform like a tensor.
Thus, the covariant
derivative
will also
8r
).
^J+
sr&-rj w sr;,-rr
r*mr
ffl
be a tensor. This allows us to write the tensor
8r; r
8^
?fc
(122)
in the simple
form
fc
(123)
-(sri*).,.
most easily proved by introducing a geodesic
0; for in this system the rightsystem of coordinates in which FJ.Z
hand sides of (121) and (123) are at once seen to be identical.
This tensor relation
If
we multiply
is
(123)
by ^(~g)g
Hence the integrand of the
lk
we
get, using (IX. 85, 86, 90),
integral in (120) has the form of
This integral can therefore be transformed to
first
an
an
ordinary divergence.
lk
integral over the surface of 2 and, as the variations 8g and their first
derivatives are zero on this surface, the first integral in (120) is zero.
Further,
we
get from (IX. 69')
and the integrand of the second
integral in (120)
becomes
XI,
THE GENERAL THEORY OF RELATIVITY
12.5
Thus we get
for the variation of the invariant J^
80'V(-flO **
Similarly,
335
we obtain by means
dx
J 2AV(-</)
125 )
<
12
of (124)
- -
**
J \g lk S/'V(-ff)
Hence, adding (125) and (126), we see that the variation of the invariant
(127)
flf)cfa
is
8J
equal to
where
37, A is
tions (12).
=-=
$y
[ Jl/^
lk
<J(g)
dx,
(128)
the tensor appearing on the left-hand side of the field equa-
The
field
equations in empty space
M =
(129)
lk
are therefore equivalent to the vanational principle
5.7
for all variations
where
and
8f/'
(130)
their first derivatives vanish at the
boundary of X.
Tt is clear that (125) and (128) remain true if from J and J we
subtract any integral whose integrand has the form of an ordinary
divergence, for this can be transformed into a surface integral which
will give
no contribution to 8^ and
the contribution to
terms
in (110)
R<J(y)
the variations in question. Now
J(g) arising from the first two
S.7 for
lk
lK
can be written
first two terms are ordinary divergences, they may thus be
lk
neglected. Further, substituting for &g /dx? the expression (IX. 68) in
terms of the Ohristoffel symbols, and using the relation
and, since the
the last two terms in (131) are easity seen to reduce to 2
j(-g)g<(v;k rirl
- r^rL).
with
(132)
FUNDAMENTAL LAWS OF GRAVITATION
336
IN
XI,
Since the contribution to R^J(-g) from the last two terms in (119)
fi,
we
is
get from (125) and (128)
8
and
fi
dx
dx
J J(~g)(R lk -\Rg lk
j *J(-g)Mlk
where
The
125
dx and
integrals J
8g*
dx
(133)
dx,
(134)
Z+2Xj(-g).
(135)
dx are not invariants. Nevertheless, since
they are defined by the same equations
(132), (135) in
every system of
coordinates, the relations (133) and (134) have a co variant meaning.
In contrast to the integrands of Ja and J, fi and
do not contain the
second derivatives of the metrical tensor. They can therefore be regarded
as functions of the quantities g lk and their first derivatives
(136)
o
As
8g]
lk
dx
,$g
we obviously have
ff
*--,(-
[8gT<
daf\d
and similarly
1
"*
Since the expressions (133) and (137) for 8
fi dx must be equal for
J
lk
any variation 8g inside the arbitrary domain S, we must have at every
In the same
way we
get
by comparison of
(134) arid (138)
In applying this argument we have treated the variables g lk g\ k as inde1
k
lk
kl
g
g\
gf
pendent, thus disregarding the symmetry relations g
This is obviously permissible, provided that the result of the derivations
,
on the right-hand sides of (139) and (140) is symmetrical in i and k.
and $ in general have rather complicated transformation
Although
the
covariant expressions on the right-hand sides of (139) and
properties,
(140) must transform like tensor densities.
XI,
THE GENERAL THEORY OF RELATIVITY
125
k all
g\
337
lk
the variables y are multiplied by a factor A,
If for constant
the quantities g lk> <J(g), and Fj^ are from (IX. 7, 4, 50) seen to be
2
l
1
multiplied by the factors A" A~ A~ respectively. Hence, the quantity
,
2 will be multiplied by the factor A~ 3 i.e. 2 is a- homogeneous function of
the g lk of degree
3. From Euler's theorem we thus get the equation
,
In the same
way we
homogeneous of degree
126.
see that
2.
as a function of the variables
Hence we
also
k
g\
is
have
of conservation of energy and momentum
was shown that the conservation of electric charge
The laws
In
115
it
is
0. This is
consequence of the covariant divergence relation div{$ }
connected with the circumstance that the vanishing of a covariant divergence of a four-vector is equivalent to the vanishing of an ordinary
divergence of a vector density. By subsequent integration over the
space coordinates we are then at once led to the conclusion that the
total charge
is
constant
in time.
The law of conservation of energy and momentum, which has the form
(X.41)or
(143)
however, not in general equivalent to the vanishing of an ordinary
divergence and will therefore not immediately give rise to any conservation laws by integration over the space coordinates. Only in the case
114 is the right-hand side of
of a stationary system considered
is,
and by subsequent integration over the space
coordinates we get a constant of the motion which may be interpreted
(143) zero for
4,
as the total energy.
The occurrence of the term on the right-hand side of (143) indicates
that the system is not strictly closed, this term being analogous to the
external four-force density on a non-closed system in the special theory
(cf. Chapter VII). In the case of electromagnetic forces, it
was possible by means of Maxwell's equations to write the four-force
of relativity
density as the divergence of the electromagnetic energy-momentum
tensor. Similarly, by virtue of the field equations (12), the term
3595.60
FUNDAMENTAL LAWS OF GRAVITATION
338
on the right-hand side of (143) can be written
IN
in a covariant
._.
XI,
way in
126
the
dx"
where
is
a scheme of 4 quantities depending on the components of the
first derivatives. The quantities k % and t* will
metric tensor and their
of course not transform like tensors, but the equations (143), (144),
(145) will, as we shall see, by integrations over the space coordinates,
give rise to conservation theorems for quantities which have simple
transformation properties and a simple physical meaning. f
In order to prove (145) we first remark that (144) can also be written
by
virtue of the rules for lowering indices together with the relations
(IX.
7).
Next we substitute from
As
the
is
sum
(12) in (144')
a function of glm and
of the
Thus we
first
two terms
see that k
is
(
tfg
and use
(140),
which gives
only and
inside the brackets
is
equal to
of the form (145) with
(146)
Using (145) in (143) the laws of conservation of energy and
can be written in the form
X*
where
448 (1918), F. Klein,
Odtt.
(148)
778 (1915), Ann. d Phys. 49, 769 (1916), Perl. Ber.,
Nachr., Math.-phys. Klaase, p 394 (1918).
t A. Einstein, Berl. Ber., p.
p.
= j(- g )(T*+t*).
momentum
THE GENERAL THEORY OF RELATIVITY
126
XI,
The
quantities
k
t
/\l(
do not
g)
339
in general transform like a tensor.
m the
y[
Since
is quadratic in
quantities t* can, if we neglect the Aterms, even be made equal to zero at any given point by introducing a
k
system of coordinates which is geodesic at this point. Z l /^(g) will
,
behave
constant.
the transformation coefficients af are
integration over the spatial coordinates we
a tensor only
like
However, by
if
can derive a set of quantities which behave like a four-vector under
a much wider group of transformations. Let us consider an isolated
system for which the TJ- are different from zero only inside a certain
extension in 4-space. For sufficiently
large space-like distances from the tube, we may assume that for a proper
choice of coordinates the y lk reduce to the constant values G lk of the
tube with a
finite space-like
special theory of relativity. Such coordinates, which can be quite
arbitrary inside the tube, will be called quasi-Galilean. If we neglect
the A-term, the Z k will therefore be zero at sufficiently large spatial
distances from our system, and by integration of (147) over the space
t
coordinates x l x 2 x*
,
we
get at once
f 3; * dx*dx*dx*
0.
(149)
dx* J
Hence we
see that the quantities
P =
X* dx*dx*dx*
W+'i dV
(150)
are constant in time.
Further, it is easily seen that the PI are invariant for any transformation of coordinates x'
x' (x k inside the tube which leaves the
1
x unchanged outside the tube. To prove this we only need to introduce
a third system of coordinates x" which on the hypersurface iij_ defined
l
by x"* = a =
constant coincides with the system (x ) and on the different
1
Since
b, coincides with the system x'
hypersurface 13 2 where x"*
the constancy of the quantities (150) holds in all three systems of
1
coordinates
On
we
get at once
the other hand, for linear orthogonal transformations
x' 1
oclx
(152)
1, the quantities Pt will transform like
|o*|
the covariant components of a four- vector in a pseudo- Cartesian system
of coordinates. For the proof, we first remark that the quantities
with a determinant
Xk
T
transform like the mixed components of a tensor under the
FUNDAMENTAL LAWS OF GRAVITATION
340
IN
126
XI,
transformations (152) with constant coefficients a*. Now consider a
four-vector a 1 whose components are constant inside the tube. The
components
= Xfa
bk
will,
on account of
(153)
and the quantities
are then also constant inside the tube,
(154)
(147), satisfy the equations
in all systems of coordinates connected
by the linear orthogonal transformations (152). Thus, formally, the situation here is the same as in
63, apart from the use of the imaginary time representation in Chapter
VI. By the same reasoning as that used for the derivation of (VI. 31),
follows from (155) that the quantity (b*lc)dx*dx 2dx* =~ l a is
invariant under the orthogonal transformations (152) and, as this must
it
now
hold for an arbitrary constant vector a 1 the integrals PI must transform
like the covariant components of a four-vector. Combining this result
,
with the fact expressed by (151) we see that the P must behave like a
four-vector by any transformation which outside the tube has the form
l
of a Lorentz transformation.
(Pt
Ejc) represent the total momentum P and
quantities
of
an
isolated
0, the expression for the total
system. If t*
energy
114 for a stationary
to
obtained
in
reduces
the expression
energy
The
E
E
system. The last term in the integrals (150) may be interpreted as the
contribution of the gravitational field to the total momentum and
112, a unique separation into a
energy. As already mentioned in
material and a gravitational part is not, however, possible. The separation depends,
general, on the coordinates used in the evaluation of
momentum.
now tempting to consider
the energy and
It is
servation laws.
The
(147) as the differential
form of the con-
quantities
(156)
which in the case of if
reduce to the expressions (X. 45, 47), will
then be interpreted as the energy flux, energy density, and momentum
THE GENERAL THEORY OF RELATIVITY
126
XI,
341
density, respectively. It should be remarked, however, that this interpretation is not independent of the coordinates used; for tj and t* and
therefore
S and
l
of space vectors
g will not in general transform like the components
by a simple change of the spatial coordinates, neither
L
k in general be an invariant under such transformations. Only the
expressions (150) for the total momentum and energy have an invariant
will
meaning which
127.
is
physically satisfactory.
Different expressions for the densities of energy
and
momentum
By means of (146) and (132) we can now write down explicit expressions for the quantities t t k As shown in Appendix 9, the derivative of
with respect to gl is given by
.
After a simple calculation
_
flU*-M.
As shown by
r*
1
Jm
we then
get,
)u-
(is?)
S
**
*.$
(15*)
(158)
remembering that
aV(-)
~~ _~ ~ W
-- -- pi. --- -,
Tolman,")" the quantities
Z k may
t
also be expressed in
the very useful form of an ordinary divergence
From
in
(148), (13),
and
(146)
we
get
which the A-term has disappeared. This expression can be further
transformed by means of (139)
t R. C. Tolman, Phys. Rev, 35, 875 (1930).
FUNDAMENTAL LAWS OF GRAVITATION
342
As shown
Hence,
in
Appendix
Xf may be
9,
the last term
is,
IN
XI,
127
however, identically zero.
written in the form (159) of an ordinary divergence
'
of the quantity
8* m
~(7
(162)
K vym
we
Further,
small
get from (148), (146), (142), and (159), neglecting the
A- terms,
-_.=
J(vv
")T'_ K~= dx m
'
ff
(163)
'
obtained from this equation in
Substituting the expression for
the expression (146) we get for t 4 * and, finally, for the 'energy density'
Vx*
from which the
total energy
is
obtained by integration over the spatial
coordinates.
The
128.
mass and
gravitational
total
energy and
momentum
an isolated system
of
By an
isolated system
we understand a system which
allows the
introduction of quasi-Galilean coordinates x l
(x, y, z, ct) in which the
line element at great distance from the system takes the form (86'),
where
m is a constant.
system we then
1
=-
For the total energy-momentum vector
get by means of (150) and (159)
4
f I, dx^dx*dx*
l
cj
! f
of the
dx*d
c] dxv
(165)
The
first
integral
integral over
an
on the right-hand side can be transformed into an
infinitely distant spherical surface
r
x *+y*+z 2 )*
constant.
It thus depends only on the values of the metric tensor and its derivatives
at large distances which are time-independent. As the vector PI is
constant for an isolated system, the last integral must also be constant;
THE GENERAL THEORY OF RELATIVITY
128
XI,
343
must have the value
zero, fcince the integral involved cannot
a
finite rate in an isolated system.
at
constant
change permanently
it
actually
Hence, we get
P^ =
/.
8^
da,
(166)
where n^ = dr/dx*1 = (xjr, y/r, z/r) is a unit vectofr in the direction of tha
outward normal, and the integral is extended over the surfac^ of the
infinitely distant sphere r
constant.
At
large distant^
V"
we
have,
however, according to (86'),
ffii
all
022
=~
= .2m
+
!
033
>
/.
2ra\
other components being zero. Hence,
S^i.SW.-IU.-UfS.
,,67,
If we neglect terms of the order
gik jyy Qik anc j
From
F^.
(162)
^(~y) \}y
and (157)
m/r compared with unity, we can replace
when multiplied by a Christoffel symbol
we therefore get to .this approximation
(168)
As
G lk
for i 7^ k the only Christoffel symbols occurring in this
are
the following, which aie easily calculated by means of
expression
=^=
and
(EX. 50)
T>
1
^4
__
(167):
x4^^
t
"2
__
ps
l
rs
/*'
_M _s
I
\~ m"
r4/
^s ~" G^ri*
^ r "~ o
(2r8 ri*
r'
(169)
Hence we get
*/*
-~8f
*^7>j
w
- --
ACT*"
and from
(166)
/>
As
PI
= -8
(P,
(170)
^ + m^ =
+
*r 2 Lw
cjf ^[('j'
\r/
E/c),
w^u
we
-8f !=?.
\r/J
see that the total
momentum
KC
is
(171)
zero in the
system of coordinates corresponding to the line element (86') at infinity,
the isolated system is as a whole at rest in this system. For the total
energy we have, on account of
(88),
E = -cP4 =
= Mc\
(172)
the gravitational mass
as determined by the scalar gravitational
potential at great distances, is connected with the total energy by
i.e.
FUNDAMENTAL LAWS OF GRAVITATION
344
IN
XI,
128
Einstein's relation. For a Lorentz transformation of the quasi-Galilean
transform like a four-vector and in the
l
P
E will have the same values as the momentum
coordinates the quantities
transformed system P and
and energy of a moving
particle in the special theory of relativity.
In the calculation of the total energy just worked out we have
used the formula (159) which expresses the energy as a function of pure
gravitational field quantities, actually only the field variables at infinity
occurring in the expression (166). Sometimes it is convenient to use
instead the formula (164). For a stationary or a quasi-stationary system
where all terms depending on time derivatives can be neglected, we get
E = -cP4 = -
X 4 4 dx
dx*dx*
In the
last integral the integration is
sphere of radius
(169),
and
constant, hence
r'
again extended over the distant
we get by means of
(162), (157),
(170)
*;/*
i
==
rs
i>rs -o^r^o^o
ir
r +8r^)-j(G
"
ir/
ro/
-{rft~i(8^rf
ID
Lf
*
-^n
**
11
=aj*,
and a comparison with (166) shows that the last integral
E. Thus we get for E the simple expression
equal to
in (173)
T\)dV,
is
(174)
which contains only an integration over the part of space where T* is
from zero, i.e. where there is actually matter present. A com-
different
parison of (174) with (172) shows that the quantity
(175)
Tl)
may
Since
be interpreted as the density of gravitational mass in the system.
/* obviously behaves like a scalar for all purely spatial transforma-
tions this interpretation has a well-defined physical meaning.
In the case of an incompressible fluid at rest, treated in 124,
according to
(92'),
T\
T\
T\
= $,
T\
-/I
c2
we have,
XI,
128
THE GENERAL THEORY OF RELATIVITY
Further, for A
and
0,
we have on account
345
of (103), (104), (102), (107),
(175),
rsn
Hence, we get for the total
field -producing
gravitational
mass of the
spherical fluid
in accordance with (108). While the proper rest mass density /S
constant in an incompressible fluid, the gravitational mass density
decreases with
according to (176).
is
IJL
XII
EXPERIMENTAL VERIFICATION OF THE
GENERAL THEORY OF RELATIVITY.
COSMOLOGICAL PROBLEMS
129. The gravitational
WHILE the consequences
shift of spectral lines
of the special theory of relativity have been
a
to
verified
very high degree of accuracy by numerous experiments,
the experimental verification of the general theory has so far been
limited to three cases only.
nected with the fact that the
The reason for this is obvious and is conNewtonian gravitational theory represents
a very good approximation for
all
phenomena
gravitational
inside the
solar system.
The most elementary of the effects which represent a crucial test of the
general theory of relativity is the gravitational shift of spectral lines,
which is a direct consequence of the principle of equivalence. According
to (VIII. 100), the rate of a standard clock at rest in a gravitational field
at a place with the scalar gravitational potential x is connected with the
rate of the coordinate clock defining the time
of our system of co-
ordinates by the formulaf
dr
Now
consider an
atom
^dt(l+2 x /c^
at rest at the place
Xi, emitting light of the proper
v.
(I)
p l with the scalar potential
Then v is equal to the num-
frequency
ber of light waves emitted per unit time in the time-scale of a local rest
system of inertia which is the same as the scale of the standard clock at
rest at the point
p r The number of waves emitted
per unit time in the
scale of the corresponding coordinate clock will then be
v1
for, according to
sponding to AJ
(1),
1
$(l+2 Xl /c),
(2)
the time-interval AT of the standard clock correis
AT
(l
+ 2^
2
1
/c
If the gravitational field
is
stationary, the (j lk are independent of t and the number of waves reaching
an observer at an arbitrary point p 2 per unit time in the t -scale must be
constant
in
time and equal to the number of waves
emitted from
f This formula, which is a consequence of the principle of equivalence, has also played
discussions between Einstein and Bohr on the consistency of the
Bohr in Albert Einstein Phtloquantum mechanical description See the article by
ftopher-fictenttst, Library of Living Philosophers, voi vn, Kvanston, 1949.
an impoitaiit part
XII,
THE GENERAL THEORY OF RELATIVITY
129
347
per unit time in the same scale. The frequency v of the radiation measured
by a standard clock at rest at p 2 will then be
v
where ^ 2
+ 2 X2 /c2)-^,
v 1 (l
(3)
the scalar potential at the place p 2 for a unit time-interval
in the scale of the standard clock at p 2 corresponds to a time-interval
From
(2)
is
and
(3)
we thus
get
(*>
Hence the observed frequency v will differ from the proper frequency v
by an amount Ay = v-~v, which in the case of weak fields is given by
where A# = Xi~~Xz * s ^e difference between the potentials at the places
where the light is emitted and observed, respectively.
In the gravitational field of the sun we have, according to (XI. 83, 88),
kM
me 2
M=
where
is
the mass of the sun.
1-983
10 33 gm.
Hence we get
(5')
for the shift of a spectral line
emitted by an atom in the outer layers of the sun as compared with the
same line emitted on the earth
^ =_*(!_.!),
rj
c2
where
i\
and
r 2 are
(6)
\r t
the r -values at the surface of the sun and at the
distance of the earth, respectively. These values of r are not exactly
equal to the values of the distances calculated by means of the usual
astronomical methods which are based on the assumption that the space
is Euclidean and that the light rays are moving in straight lines. The
differences are small, however,
and
since the effect (6)
is itself
small, the
by using the astronomical values of the distances
rly
instead of the correct r- values is small of a higher order. Since r 2
error introduced
we get, putting
ofthesun
>
the value
rl
equal to the usual astronomical value of the radius
^=
^r
v
6-96x10"
2 12
'
cm.,
10~ 6
(6')
(7)
EXPERIMENTAL VERIFICATION OF
348
129
XII,
The observed spectral lines should therefore be shifted
the
red. This interesting effect predicted by Einstein!
towards
slightly
is in satisfactory agreement with the observations both in the case of the
sun and in the case of the heavy companion of Sirius, where the effect is
about thirty times larger 4
for the line shift.
The advance of the perihelion of Mercury
Let us consider the motion of a particle (a planet) in the gravitational
field of a much heavier body (the sun) described by Schwarzschild's
130.
exterior solution (XI. 83, 88, 24). In this case, the components of the
metric tensor are
ffn
g lk
22
>
,-
A f
for
2
**
0r 33
r 2 sin 2 0,
gu
= -/ 1
^J
^
,
k,
2kM
K Mc*
47T
(8)
Hence
Yi
>
Yuc
According to (X.
-"2
the gravitational force on the particle
17, 17'),
is
(io)
where
is
m=*
the relativistic mass of the planet. The square of the velocity
is
r2
(11)
where the dots mean differentiation with respect to the time-variable
Further, by (X. 12) and (8), the momentum vector of the particle is
The equations of motion (X.
15)
_
dt
t.
dt
2 dx l
'
35 r 898 (1911).
John, Astrophy*. Journ. 67, 195 (1928); W. Adams, Proc. Nat. Acad. 11, 382
(1925); see also E. F. Fjeundlich and W. Ledermann, Mon. Not. Rvy. Astr. Soc. 104,
1, 40 (1944).
t A. Einstein,. Ann. d. Phys.
I St.
THE GENERAL THEORY OF RELATIVITY
130
XII,
349
are thus in the present case
(146)
Guv
and
~(rr*sinWd>)
(He)
0.
at
Since the gravitational field is static, the energy
by (X. 22) is constant. Hence
is
first
H of the particle defined
where
integral of the equations of motion,
is
a constant of
integration representing the energy of the system divided
From (146)
we see that
^
'
01
by
?fe
c2
represents another integral of the equations of motion. On account of
the central symmetry of our problem, any plane through the centre may,
however, be chosen as the plane 9
|TT, i.e. the orbit can lie in any plane
the
from
centre.
(16) in (14c) we get at once the
through
Substituting
^_^
further integral
where
(17)
On
another constant of integration.
equation can also be written
is
account of (15) this
(7.
(18)
loc/r
Now
a/r
is
even for Mercury, the planet nearest to the sun, the quantity
a very small quantity of the order of magnitude
OL
and
for the other planets a/r is
cases, the gravitational field
also
iJjJL
u z /c 2
<^
c 2r
much
may
/-v
5X10- 8
/ t
smaller.
f\\
(19)
Therefore, in
be treated as a weak
field
all
actual
and, since
we may
to a first approximation use the approximation
Further, using (11) and (16) we therefore get for
in (15).
(X. 25) for
(15) to this approximation
constant,
(20)
EXPERIMENTAL VERIFICATION OF
350
To the same
the usual energy equation in Newton's theory.
approximation, (18) reduces to
which
is
r 2 <f>
C,
(21)
expressing the conservation of angular momentum.
mined by (20) and (21) is therefore an ellipse
where
orbit deter-
/22)
the eccentricity, and
is
are the
The
_?
130
XII,
r- values
corresponding to aphelion and perihelion of the planet,
respectively. For Mercury we have
e
For stronger
fields
0-2056,
5-786
10 12 cm.
(24)
the equation (20) has to be replaced by the energy
(21) we have the integral (18). However,
and instead of
equation (15),
the left-hand side of (18) cannot in general be interpreted as angular
momentum, since the notion of a 'radius vector' occurring in the definition of the angular momentum has an unambiguous meaning only in a
Euclidean space.
In order to determine the orbit
l/r instead
quantity p
of
r.
m the general case,
Thus we
drdCI
dr
a\
-
and by means of
(11), (16), (18),
particle
*
Using the expression
- G(
(
and
r/r/
0) for
-acp)
we introduce the
get from (18)
= - ldp\n/-i(l
(25)
we
-"p)
/or\
(28)
'
get for the velocity of the
\2
+p
-/>
(26)
F and squaring the energy equation
(15)
we
get
(27)
2
Thus, substituting the expression (26) for u and solving with respect to
(dp/dffi, we get the differential equation for the orbit of the particle in
the form
where
equation
7/7x2
and
is
= Al
are constants.
On
Bp-p'+ap*.
(28)
account of (19) the last term in this
2
very small compared with the term p and,
if
we
neglect
it
XII,
THE GENERAL THEORY OF RELATIVITY
130
entirely, (28) reduces to the equation obtained
equations
(20), (21),
351
from the Newtonian
e.
'
(2
>
approximate equation. The
values of p corresponding to perihelion and
aphelion are obtained as roots of the quadratic equation
It is easily seen that (22) is a solution of this
maximum and minimum
A + Bp-p* =
Q.
(30)
Let p l and p 2 be the two roots which must be real and positive in the case
of an orbit which stays inside a finite region in space. Then we have,
according to
The equation
(23),
may now
(29)
be written
(32)
p 1 )(p t -p)},
and the increase
</> 2
cf)
in
<f>
during an increase of p from p l to p 2
is
obtained by integration
P2
-
,-*-J
sin-
(33)
Pl)
2(P2
-P)}
Pl
in
accordance with the solution
(22).
Returning now to the exact equation
(28), we see that the right-hand
a polynomial of third degree and the equation obtained by
has three roots p lt p 2 p 3 For small values of a two
putting dp/d(f)
of these roots, p l and p 2 say, must be approximately equal to the two
side of (28)
is
roots (31) of the equation (30),
and since
Pi+P2+Pa
34 )
must be very large for small values of a. The roots p 1 and p 2 will therefore again represent the minimum and maximum values of p in the orbit
p3
of the planet. Instead of (32)
-^
VHP
PI)(P
we now have
P 2 )(P
Pa)}
where we have used the equation
a (Pl+P2)JI
EXPERIMENTAL VERIFICATION OF
352
following from (34). Since ocp and
get to a first approximation in OL
cx(/o 1
+p 2
XII,
are small quantities
130
we thus
+ *p\
2
and the
increase in
<f>
for
an increase of p from p i to
/> 2
is
PI
'
X
Hence the
difference in
<f>
between two successive perihelions
is
(36)
which
differs
from the value
2-77
obtained from (33) by the amount
A<=~(p
+? 2
(37)
).
be interpreted by assigning to each revolution of the
planet an advance of its perihelion of the amount (37). Since it is a small
effect we can substitute the approximate values (31) for p l and /> 2 F r
This result
may
Mercury we
an advance of the perihelion of an angle
get, using
42'9" per century, which is in satisfactory agreement with the observed
advance after subtraction of the effect due to the perturbation by the
other planets. f For the other planets the advance of perihelion following
from Einstein's theory of gravitation is too small to be observed with
(24),
certainty.
Instead of the Schwarzschild form (XI. 83) of the line element we
could in these calculations also have used the form (XI. 86) corresponding to the 'isotropic' coordinates (XI. 85). In these coordinates the
equations of motion are somewhat more complicated, but the final
result regarding the advance of perihelion is, of course, the same
a&that given by
(37).
t J. Chazy,
CM.
182, 1134 (1926).
THE GENERAL THEORY OF RELATIVITY
XII,
131
131.
The
The
353
gravitational deflexion of light
which furnishes a
crucial test of the general
theory of relativity is the deflexion of a light ray in the gravitational
field of the sun. Since this field is static, the trajectory of a light ray is,
according to the considerations of 117, determined by Fermat's principle,
third of the effects
by the equations (X.
e.
dX"d\
100, 99, 94):
>~~
lK
~'
a-
In the
field
described
by
and
(8)
(9)
(38)
1-
we thus have
(40)
and the two equations
~
d\
From
(42)
(r*6)
we
V
"
(38) with
2r 2 sin
cos
reduce to
2
-- 0,
sin 2
-f- (r
d\
0)^=0.
(42)
see again that
B
are integrals
-~ 2, 3
and
(41)
r*</>
(43)
then reduces to
a\
2/1
2
1/r as
dX
r2
C*a
r/
Introducing p
=,
\TT,
I4A\
44
=--.
2
c
a new variable in (44) we get, since
d<f>
d\
2
where we have put
A K\
cC
Now
consider a light ray coming from infinity (p
0) along the
3
direction
0. Neglecting first the small term ap in (45), this equation
</>
is
easily integrated.
We
get
p
i.e.
3595.60
/>
= --sm<,
Aa
=-
-.
(46)
EXPERIMENTAL VERIFICATION OF
354
If
we make a
coordinates
r,
picture of the trajectory in a Euclidean plane, where the
are pictured as ordinary polar coordinates, the curve
<f>
which passes the centre
and which goes to infinity again for
(46) represents a straight line
->
for
<f>
\TT
131
XII,
</>
The exact equation
(45)
may now
at a distance
TT.
be written
where we have introduced a new variable
a
instead of p. Since ap
is
in a
Ap(l
(48)
ap)*
a small quantity,
/v /"
we may write
/vnl
to the
first
order
rurr\
(49)
Hence
and from
dp
dff(l
+~
(47)
According to (47) the maximum value p m of p, i.e. the value for closest
approach of the ray to the sun, corresponds to o- = 1. The corresponding
value of the angle
d>
is
Thus, in the Euclidean picture mentioned above, the curve (50)
represented by a slightly curved line with a total deflexion
is
This will be the deflexion of a light ray in the field of the sun as noted
by an observer on the earth, which in this connexion may be regarded as
infinitely far
from the sun. From
(49)
we get for the p- value correspond-
ing to closest approach
Hence, neglecting terms of second order in
a,
we
get for (52)
(53)
For a light ray which grazes the limb of the sun we then get, by means
of (5') and (6'), an angle of deflexion of 1*1 5". This effect predicted by
THE GENERAL THEORY OF RELATIVITY
131
XII,
355
Einsteinf has been tested by observations during total eclipses of the
sun on the apparent positions of stars whose light has passed close to the
limb of the sun. The agreement between Einstein's formula (52) and
the observations seems to be satisfactory, J but, as the effect is just
inside the limits of experimental error, one cannot attach too much
weight to the quantitative agreement.
The
deflexion (52)
pressed
by
(40)
is
due partly to the varying velocity of
and partly to the non-Euclidean character of the
we had taken the space
would have been replaced by
geometry. If
(44)
spatial
to be Euclidean, the equation
(54)
c (l
Instead of (45)
light ex-
ct/r)
we would then have obtained
df)
which by integration to the
first
order in a leads to a deflexion
i/j
a/A
(55)
of only half of the value (52).
On the other hand, if the velocity of light
is put equal to the constant c,
while the spatial geometry is taken to be that given by the metric tensor
y iK defined by (9) and (8), the trajectory of the light ray is, according to
(38), a 'straightest' line. In this case the equations (44) and (45) are
replaced by
respectively.
deflexion
(f -"-">(*-
1oL/r
Integration of the last equation then again leads to a
/
ifj
first order in a. Adding the two
back to Einstein's formula (52).
to the
The
^A\
/A
a //\
effects (55)
(56)
and
(56)
we thus come
direct experimental verifications of the general theory of rela-
should be remembered, however, that the
general theory is not only a natural, but a nearly cogent generalization of
the experimentally well-founded special theory. Further, since Einstein's
tivity are thus very few;
it
gravitational theory contains Newton's theory as a
tion,
all
predictions of Newton's theory
t A. Einstein,
J
first
approxima-
the numerous astronomical observations which confirm the
BerL Ber.,
W. W. Campbell and
may
therefore in a certain sense also
Ann. d. Phys. 49, 769, 22 (1916).
Trumpler, Lick Observ. Bull. 11, 41 (1923); 13, 130 (1928).
p. 831 (1915);
EXPERIMENTAL VERIFICATION OF
356
131
XII,
be regarded as a support of the general theory of relativity. The fact
that the differences between the two theories become apparent only in
the three small effects discussed above simply shows that Newton's
theory represents an extremely good approximation for all gravitational
inside the solar system. However, as regards cosmological
which
concern the structure and motion of larger parts of the
problems
phenomena
two theories of gravitation can be expected to lead to very
results and, on account of its inner consistency and the gener-
universe, the
different
ality of its basic principles, Einstein's gravitational
theory
may
be
expected to provide a safer guidance in the handling of these difficult
problems.
132.
It
Cosmological models
has been
known
for a long
theory meets with serious
timef that Newton's gravitational
when applied to the universe as a
difficulties
Since Einstein's gravitational theory can be expected to give
results which deviate appreciably from Newton's theory just for systems
whole.
of cosmological extension it is clearly of great interest to investigate the
possibilities for a treatment of the universe as a whole, offered by
the general theory of relativity. This question was taken up by Einstein J
new
shortly after the development of the general theory and has since then
been the object of many investigations by numerous authors.
shall
We
not attempt here to give a detailed account of all these investigations,
but confine ourselves to the consideration of the static homogeneous
models of the universe originally proposed by Einstein and by de Sitter.
All models of the universe which have been considered so far are
based on the assumption that the world, when looked at from a large||
scale point of view,
is
spatially
homogeneous and
isotropic.
It
is
true
is partly gathered into stars which again
have a tendency to cluster into nebulae of the same character as our own
galaxy of stars. But, in the portion of space which can be reached by
means of the present telescopes, these nebulae seem on the whole to be
fairly uniformly distributed, and the assumption that the large-scale
properties of the universe can be properly described by treating the
matter as a perfect homogeneous fluid seems to be a natural startingpoint. In the models proposed by Einstein and de Sitter the universe is
that the matter in the universe
t C. Neumann, Kgl sacks
H. v Seehger, Astron. Nachi.
t A. Einstein, Berl Ber p.
A Einstein, Serf Ber., p
,
||
W. de
Sitter,
Ge# d Wiss zu Leipzig, Math.-nat. KL 26, 97 (1874),
137, p. 129 (1895), Munch. Ber. 26, 373 (1896).
142 (1917), Ann. d. Phys 55, 241 (1918)
142(1917).
Amst. Proc. 19, 1217 (1917); 20, 229 (1917).
XII,
THE GENERAL THEORY OF RELATIVITY
132
357
system, which means that we can
introduce a system of coordinates x -^ (r O,<f>,ct) in which the line
element has the static and spherically symmetric form (XI. 63)
furthermore assumed to be a
static
rf*
where a and
-- a(r)
(J/-
+r
(rf0M-
d<f>*)-b(r)c* dt*
(57)
On account of the assumed homomay be taken as the origin
b are functions of r only.
geneity of the universe any point in space
r =of the spatial system of coordinates
The functions a(r) and b(r) are now connected with the proper mass
density /I and the proper pressure j8 in the universe by the field equations
(XI. 96, 94) for a perfect fluid,
'+
0,
(58)
--abr
r*
ar
Here, p and /t are constants on account of the assumed homogeneity of
the model and we shall look for the possible regular solutions of those
equations.
Since dp/dt
0, (58)
reduces to
2
(/ic
which requires either
b'
or
0,
(60)
(61)
/ir
These two alternatives lead to
-/})//
(62)
fj
tlie
solutions of Einstein
and de
Sitter,
respectively .f
133.
The Einstein universe
The
by the condition (61) corresponds
and by a suitable choice of the time variable
solution characterized
constant value of
constant
may
fe,
be made equal to
to a
/
this
6-1.
(63)
Introducing (61) into (59 a) arid solving this equation with respect to a,
,
,
we obtain
~
a
(64)
2
'
R. C Tolman, Proc Nat Acad 15, 297 (1929)
EXPERIMENTAL VERIFICATION OF
358
where we have introduced a new constant
E by
XII,
133
the equation
(65)
Further, using (64) and (596),
-p
which together with
we
get
l*(/i
e+$),
(66)
(65) leads to the relation
i*(Ac +3$).
(67)
Now
the quantity /3, is essentially positive and even if we allow for
possible cohesive forces in the fluid giving rise to a negative value of ft,
the order of magnitude of p will for any reasonable properties of the
matter be far below the value /lc 2 Hence the constants A and l/E 2
must also be positive and B will therefore be a real quantity of the dimension of a length.
.
In his original paper Einstein assumed the matter in the universe to
be incoherent, thus exerting no pressure at all. In this case, we get from
(66)
and
(67)
^i-T*'On
the other hand,
radiation,
we
if
the universe
is
(68)
assumed
to be mainly filled with
have, according to (VII. 145),
which leads to the relation
>
A==
2^ = ^ C
^,,0/2 '
f7{\\
(70)
According to the estimate of Hubble,f the lower limit for the mean
density of matter in the actual universe is of the order
fr
From
(68)
and (XI.
24)
:=
tt 10- 30 gm./cm. 3
we then
_L~
and, as an upper limit for E,
3
f
get
9xlO~ 58 cm.~ 2
(71)
we obtain
1C 28 cm.
X 10 10
light years
Hubble, Astrophys. Journ. 79, 8 (1934).
XII,
THE GENERAL THEORY OF RELATIVITY
133
According to
universe
and
(57), (63),
is
359
element of the Einstein
(64) the line
In a region of space for which
(73)
^<!>
this line
element reduces to that of the special theory of relativity.
With the value
of A and
R given by
we get for the distance r
(71)
tt IO 14
of the planet Neptune
T2
:__
/]0 14 \ 2
[
io- 28
which means that the condition
(73) is
amply
satisfied inside the solar
system. This gives the justification for neglecting the A-term in the
treatment of all gravitational phenomena connected with the motion
of the planets Furthermore, we see that the line element of the special
theory of relativity, which is experimentally known to hold in every
system of inertia far away from gravitating bodies, appears in the
Einstein model as a consequence of the gravitational field equations for a
homogeneous static distribution of the distant celestial bodies.
For larger parts of the universe, where the condition (73) is not satisfied, we get from (72) for the spatial metric tensor and the dynamical
gravitational potentials
rii
72/
r2
#2'
r2
^33
'
r 2 sin 2 <9
7 IK
= n*for ^
*
y,
The
real only for r
0,
bur
r 4 sin 2
},
'
(74)
Y^
0.
(75)
clement
spatial line
is
*>
<
&
(76)
R, which defines the extension of the physical space
The total volume of the universe is
in the Einstein universe.
TT
27T
IT
27T
r 2 sinedrd6d<t>
000
000
EXPERIMENTAL VERIFICATION OF
360
and the greatest distance from the
obviously
if
is
origin
~-
=.
Further, since the matter
on account of (X. 56)
mass
dV
fi
dV
2~ E
and
(68) or (70)
we
(78)
we
get
in the universe
--- 7r 2
and the mean distance of the matter from the
of (79)
-_
at rest in our frame of reference,
is
/3,
/?sin
for the total proper
By means
133
XII,
/P/l
origin
is
(79)
of the order
'
get
~~xl.
l,
(80)
477X1
These relations are in accordance with the relations (XI. 49, 50) suggested
by the considerations in 120 on the nature of the centrifugal and Conohs
forces
The
kind.
spatial geometry defined by (74)-(79) is of the so-called elliptical
However, the line element (76) allows also a somewhat different
interpretation as regards the geometry of the universe as a whole.
we define four new variables ?/ y lt ?/ 2 ?/ 3 by
,
If
?/
(81)
r sin 6 sin
?/ 3
where
?/o+2/i +#!+#!
</>
=^
(82)
the hne element (76) takes the Euclidean form
dv*
dyl
dyl+dyl+dyl
(83)
This shows that the physical space of the Einstein universe may also be
interpreted as the three-dimensional surface of the sphere (82) of radius
R in a four-dimensional Euclidean space with the
Defining polar coordinates R,
2/
~ R cos
J/T,
6,
<f>
on the sphere
yl
iff,
Cartesian coordinates
R sin
ifj
(82)
by
cos 9
(84)
\\e
get
Rsiinfj
(85)
THE GENERAL THEORY OF RELATIVITY
133
XII,
and to each point on the sphere corresponds one
variables
(0, 9,
</>)
<
By
set of values of the
in the intervals
l/f
^ <
77,
0<0<27T.
77,
(86)
using (84) in (76) the line element assumes the form
= #
da 2
and since
(d^ +sin
d0 2 +sinV sin 2
(87)
d<j>*),
in these coordinates
we
361
\y LK
_R 6 sin 4 </fsin 2 0,
get for the total volume of the universe
7T
7T
TT
7T
= 477#
J ^ ^snvtysinfl
$
j
000
dif>
d0
Thus the volume
J sinV
in this so-called spherical space
in the corresponding elliptical space.
(86)
which shows that the
This
is
is
^ ^ 2^
3
-
(88)
twice the volume
also clear
from
(85)
and
elliptical space covers only the hemisphere
on
|TT or, in other words, antipodal points
corresponding to
the sphere are counted as one point only in elliptical space.
Similarly we find from (87) that the total distance around the closed
</r
spherical Einstein universe
is
L = 2R
difj
(89)
277.R,
which thus represents the distance one would have to travel along a
'major circle' on the sphere (82) in order to return to the starting-point.
The relations (80) remain true also m the spherical Einstein universe.
Since the dynamical potentials (y %) are zero, the gravitational force
on a particle is zero and since, further, the system of reference is rigid,
t
a free particle in
the universe is
moving with constant
velocity along the
straiyhtest line compatible with the spatial geometry of the universe (see
end of
110)
In this generalized form the law of inertia
also in the Einstein universe.
remain at
is
thus valid
free particle at rest will therefore
rest.
According to (X. 94) and (75) the velocity of light is constant and
equal to c. Light emitted by a star at rest at the point (r, 6, </>), at the
time ^ will therefore arrive at the origin r =
at the time
EXPERIMENTAL VERIFICATION OF
362
and
since r x
is
constant,
we
by
get
A/ 2
Thus the time-interval A
133
XII,
differentiation
---
A^.
(91)
between the arrival of two successive wave
crests at the origin is equal to the interval A^ between their emission.
Further, since x
0, the time variable t is identical with the time shown
by standard
and
clocks at rest
means that the frequency
(91) therefore
of the light as determined by an observer at the origin
proper frequency of the light emitted by a star at rest.
Apart from small Doppler
of the distant nebulae
effects
is
equal to the
due to individual random motions
we should
therefore not expect any systematic
shift of the spectral lines emitted by the nebulae. In the actual universe,
however, the work of Hubble and Humasonf shows a definite red shift
from nebulae, which increases linearly with the distance.
This clearly shows that the Einstein model, in spite of its many satisfactory features, represents only an approximate description of the
in the light
actual universe.
134.
The de
Sitter universe
Besides the solution corresponding to the condition (61) which leads
model there exists another static homogeneous and
to the Einstein
isotropic solution of the general field equations arising
dition (62), viz.
0c2+
If
we add the equations
(59 a)
f
(ab)
or
--=
ab
By
and (596) we get
K (^c 2 +p)a 2 br
from the con(92)
in this case
constant.
a trivial change of scale of the time variable this constant can of
made equal to 1, which means that b is the reciprocal
course always be
ofa:
Introducing y
ab
I/a as a
(yr)'
new
1.
(93)
now be
variable, (596) can
= y'r+y
written
- +l-(
which by integration gives
yr
C
u
:r 3 +constant.
,
f E. Hubble, Proc. Nat. Acad. 15, 168 (1929), E.
Astrophya. Journ. 74, 43 (1931).
(94)
Hubble and M. L. Humason,
THE GENERAL THEORY OF RELATIVITY
134
XII,
Since y
(94)
is
must be
regular for r
where we have put
Using
the form
(95)
in
the constant on the right-hand side of
Thus we get from
zero.
-^
./I
(57)
363
(93)
and
-~
(94)
(96)
we then
get the de Sitter line element in
- MB*
<
97 >
The components of the
spatial metric tensor are again given by (74),
the spatial geometry in the de Sitter universe is of the same form as
in the Einstein model, at least if A is assumed to be positive. However,
the dynamical potentials are not zero in this case; instead we have
i.e.
Thus a
free particle is acted
0,
=~
a-
upon by a gravitational
K = -mgradx =
h55'
98 )
force
>l
(")
proportional to the variable r, which means that the law of inertia does
not hold over large regions of space in the de Sitter world. Only in regions
for
which
element (97) again reduce to the line element of the special
of
relativity and the law of inertia is approximately valid.
theory
The equations of motion of a free particle may now be obtained from
will the line
(X.
by
15)
(99).
by using the expression for the gravitational force given
is, however, more convenient to make use of the possibility
It
113 of eliminating the dynamical potentials by the introduction of a suitable set of space-time coordinates. This is attained by
discussed in
defining
new
variables
r', 0',
<f>',
t'
by the transformation
(.00,
e,
$=
<f>
EXPERIMENTAL VERIFICATION OF
364
XII,
As shown independently by Lemaitref and by Robertson,^
134
this trans-
formation leads to the following form for the line elemented 2 ==
which is easily
coordinates x
e'W(d/ 2 + r' 2
verified
',
y'', z'
by
2
2
df )-c dt' 2
d0' 2 +r' 2 sin 2 0'
direct calculation. Finally, defining
connected with r\
9',
<f>'
(101)
new space -
by the usual equations
connecting Cartesian coordinates and polar coordinates in a Euclidean
space, (101) may be written
ds 2 -= e 2d 'W(dx' 2 +dy' 2 +dz' 2 )-c 2
The new coordinates #', y' z', t' can take
With these coordinates we have
all
722
-^ -
733
- ~& _
(102)
values from
__
dt'
~oo
to
-f<x>-
^33
(103)
Vt
and the
spatial Christoffel
Thus, the time variable
rest at
any reference
'
symbols are
is
t'
At any
point.
all zero.
the time
shown by a standard clock
fixed time
if
the spatial geometry
at
is
Euclidean, x', y' z' being Cartesian coordinates apart from the common
to a point (/,#',(/>'),
factor (WIK. The distance from the origin r'
as measured by standard measuring-rods, is
',
l^
The
velocity of light
is
p/r/V.
(104 )
constant and equal to
d(J
11
c.
MO^
__ c
(10o)
d7'Further,
it is
easily seen that the trajectories of light rays are straight lines.
This follows at once from the
first
of a light ray. Taking
i -=.
equations (VITT 87) for the time track
1, 2,
3 in these equations,
we
get
and by integration
x
--
aA
--=
a'e -**'"',
(106)
where the a are constants of integration. Hence
L
dv'/dt'
dy'ldt'
dz'/dt'
(107)
E Lomaitre, J. Math and P/??y,9 (MIT), 4, 188 (1925)
H. P. Robertson, Phil Mag 5, 835 (1928)
f G.
J
134
XII,
THE GENERAL THEORY OF RELATIVITY
and the equations of the trajectory of a
of the form
x
light ray are linear equations
^o __
~~
yp __ z
~ zo
,
(IDS)
a3
a2
a1
365
Since the light rays are rectilinear in the (x y z')-space we can obviously
apply the usual tnangulation method to determine the parallax of
f
and
celestial bodies
in this
way determine
the distance (104)
by
direct
measurements.
Let us now consider the propagation of a light ray along the
From
(105)
and
(103)
we
^-fcce-cr/*.
time
/Q
-axis.
get
or
The motion of a
a:'
(110)
light signal starting off at the point
in the direction of the negative #'-axis
is
x'Q
>
at the
thus given by
(111)
and we
XQ
see that, unless
this signal will
<
Re'***'
never reach the origin
x'
1
*,
0.
Although the world corresponding to the line element (102) is apparently infinite, an observer at the Origin will never be able to obtain any
information about the regions outside the 'horizon' defined by (112).
The greatest distance inside the observable world of an observer placed
at the origin at the time
Q is
t'
the same for
thus,
by
(104)
J.IL,
V>cto/RJ?e-cta/R
and
T>
Jt,
(112),
1 O\
(1li0
times in accordance with the intrinsically static
character of the de Sitter universe. Further, since (102) is invariant
e. it is
all
against any displacement of the origin, this consideration holds for any
observer at rest in the system of reference considered here. However,
the position of the horizon will of course be different for the different
'equivalent' observers.
is
In the present coordinates the gravitational force on a free particle
and the equations of motion are given by (X. 20'):
zero
^=0,
(114)
EXPERIMENTAL VERIFICATION OF
366
XII,
134
where
= mu
p.
W>
:==z
",
(115)
dx'
Further, since the spatial Christoffel symbols are zero in the present
case,
we have
which shows that the co variant components of the
constant:
momentum vector are
^ = constant
(116)
However, the contravariant components are not constant, since we have
=
= e-^R
y *P K
(117)
in accordance with the differential equations (X. 19').
now
get
dx
dx'fdt'
61
6 2 6 3 are constants.
,
straight lines described
by
"'
"
62
dz'jdt'
~~P~
Thus the
i/
/!/
y ~~y*
ft
we
orbits of free particles are also
equations of the form
"'
(i!7)
_
_
~ dy'ldt' ~
~&i~~~
where
From
__
f*
z
~Z
*r
nAA7
i9\
9
'
but the velocity of the particle is in general not constant. Only if the
velocity at one time is zero relative to the system of reference considered here will
it
remain zero; for from (118) we see that
and dx' L /dt' will then be zero for all times.
requires p^
constant may thus be represented
Any reference point (x' ,y' ,z')
by a
freely falling particle in accordance with the
remarks at the end
Since the system of reference is not rigid, the distance from
the origin of a point with constant values of the spatial coordinates will,
of
113.
however, depend on the time according to (104). Hence, if we assume
that the nebulae may be treated as free particles at rest in the present
THE GENERAL THEORY OF RELATIVITY
134
XII,
367
system of coordinates, their distance from the origin increases with a
radial velocity
jj
dL
which
^dt'
(s
M'ltf
R^ ^R
r
/_ df*\
1'
(120)
proportional to the distance I.
This radial velocity of the nebulae must be expected to give rise to a
shift towards the red of the spectral lines emitted by the nebulae. To
is
investigate this question we consider light emitted by a particle which
is permanently located at the reference point (r', 6', </>'). According to
and
(105)
(103) the radial velocity of light
is
-'/*.
(121)
and t'2 are the times for the emission of radiation from the
and its reception at the origin r'
0, respectively, we get by
if t[
Thus,
particle
integration
o
**
dr'
On
differentiation
of two successive
-c
A^ - A
since
t'
is
R(e-^ R -e-^ R ).
(122)
we get for the time -interval A 2 between the arrival
wave crests at the origin and the time-interval A^
between their emission the following
Now,
or
e-*'/* dt'
relation:
e^-W'*.
(123)
the time shown by a standard clock at rest and since the
we have for the proper wave-length A
velocity of light is c everywhere,
of the emitted light
^
./
Similarly the wave-length as measured by an observer at the origin will be
Hence we get from
A4.
(123)
X*e&-*V*.
(124)
Further, the distance to the particle at the time of observation of the
light
is,
according to (104) and (122),
I
From
Thus,
(124)
if
we
f '0VR
R(4*t't-4yR-\Y
(125)
therefore get
we assume
AA
A-A
(126)
that the nebulae, apart from small individual veloci-
ties are at rest in the system of coordinates used here, we get an explanation
,
EXPERIMENTAL VERIFICATION OF
368
XII,
1IU
Humason To obtain quantimust have the value
of the actual red shift observed by Huble and
tative agreement the constant
R~
R in
l-66xl0 27 cm.
(126)
1-75
This value for the 'radius' of the universe
value
(7
10 light years.
is
(127)
somewhat smaller than the
in the case of the Einstein universe. It is true that this explana-
tion rests essentially on the assumption that the nebulae in the mean are
at rest relative to the system of reference defined by the coordinates
1
(x
',
y',z',
t').
This assumption,
known as Weyl's hypothesis, f is, however,
very natural and has the attractive feature that all nebulae in this model
same footing so that an observer at any other reference point
are on the
would observe the same red
shift of the light
coming from the nebulae
as the observer at the arbitrarily chosen origin of our system of coordinates.
In the original system of coordinates the motion of the nebulae is
1 00)
by solving the first equation with respect to r. Hence
obtained from
where
a constant for each nebula. Thus
r' is
according to the
at the point r
for t~>
we
see that the nebulae
Weyl hypothesis are freely falling particles which start
at t
oo and end at the antipodal point r
=R
+00.
Finally
introducing
equations
,
variables
five
z^
(z Q ,z v z 2 z 3 ,z 4 )
,
the
by
=
(129)
we have
z*
=^
(130)
^-o
and the
line
element takes the form
* = t (*>)
/i-O
8
8
-
131
The space-time continuum of the de Sitter universe may thus be pictured
as the four-dimensional surface of a sphere of radius
R in
a five-dimen-
sional pseudo-Euclidean space just as the 4-space of the Einstein
t
H. Weyl, Phys. ZS. 24, 230 (1923), Phil Mag. 9, 936
(1930).
THE GENERAL THEORY OF RELATIVITY
134
XII,
369
universe, according to (72), (76), and (83), can be pictured as a cylinder
with spherical cross-section and the axis in the direction of the time-axis.
Since the equations (130) and (131) are form -invariant under the group
of five-dimensional orthogonal transformations of the variables (z^),
the line element (102) will be form-invariant under the group L of the
l
(x',y',z',ct ) concorresponding transformations of the variables x
nected with (z ,...,z 4 ) by the equations (129). The transformations of
L thus connect systems of coordinates (x
which are equivalent
)
These transformations play the same role in the
de Sitter universe as the inhomogeneous Lorentz transformations in
the group
in the sense of
the
flat
121.
space of the special theory of relativity. f
Although the de Sitter model leads to a natural explanation of the
observed red shift of spectral lines, this model can hardly be regarded
as a satisfactory model of the actual universe for the following reason. According to the condition (92) underlying this model, the density
and pressure of the celestial matter which give rise to the line element
(97) satisfy the
Since /tc 2 :>
Even
large
equation
0, (1*32)
if
o 2
+^ ^
132 )
leads to the assumption that p is negative and very
the existence of cohesive forces in the ideal
we permit
our model, a value of p of the order of /tc 2 would, however,
be quite incompatible with the properties of any known material. Hence
(132) can be satisfied only if we take the density to be zero or at least
fluid filling
very
much
smaller than the
mean
density of the actual celestial matter.
Hence the de Sitter model corresponds to an empty universe containing
no appreciable amount of matter and radiation, the stars and nebulae
being treated then as a kind of test bodies which do not contribute
essentially to the gravitational field. This point of view is, however, not
agreement with the basic ideas underlying the general theory of
in
relativity according to which the centrifugal forces and Ooriolis forces,
for instance, are due to the motion of the distant celestial bodies relative
to the rotating system.
While the non-permanent gravitational
fields
can be explained along these lines in the Einstein model of the universe,
the empty de Sitter model does not of course afford any basis for such
explanation,, the non-permanent fields being here of the same nature as
the fictitious forces in Newton's theory.
These considerations make it probable that the Einstein model repre-
sents a better approximation to the actual universe than the de Sitter
t See rof
(1941)
3595 60
Chap XII,
p. 364,
and
C. Moller,
Dan. Mat. Fys, Medd.
18,
No.
6, p.
40
370
THE GENERAL THEORY OF RELATIVITY
XII,
134
model despite its failure to explain the systematic red shift of the light
coming from the nebulae. If we want to construct a model in which the
advantages of the two static models of Einstein and de Sitter are combined we must obviously have recourse to non-static models in which
the metric tensor is intrinsically time-dependent. Such models have
been extensively studied in the literature. f These investigations include
models which expand from an originally static state as well as models
which undergo successive expansions and contractions. Our present
knowledge of the actual universe, which only covers a limited region in
space and time, is, however, totally insufficient and thus no unique
choice between the different non-static models is possible.
f See the extensive treatment of these problems in Tolman'b book, Relatwity, Thermodynamics, and Cosmology, Oxford, 1934
APPENDIXES
1. Gauss's theorem
LET V be a finite domain bounded by a
Euclidean spaco arid
\->
lot/(j: 1 , x.z , ^3)
closed surface a iri a three-dimensional
be a given function of the Cartesian coordinates
~- dr l dv 2 cfa* 3 over the region V.
T 3 inside K. Consider the volume integral
2*2*
&XL
Assuming that the surface o- is convex so that a straight line parallel to the a^-axis
two points only, this
with constant values of x 2 and x 3 intersects the surface a
integral may, by partial integration with respect to the variable x l9 be written
(1)
a,
where the integration on the right-hand side is extended o\ er the projection a
of the sin face a on the (.r 2 o: 3 -plane, and/ + arid/" are the values off at the two
points p+ and p~ on a whose projections on the (*t 2 2* 3 )- plane he insido the face
element dv 2 dr 3 of cjj Now let da d<j~~ be the surface elements of cr at p+ and p~,
respectively, whose projections on the (.r 2 ^ 3 )-plane are equal to the element dx 2 dx^.
Then, if n+
(nj w a nj and n~
(nf /j,j, n^ are unit vectors in the direction
of the outward normals of dcr + and da~~ and if .r^ > .rf we obviously have
)
nida~-^dc 2 dx 3
nj da+
Hcnoe
(1)
may
(2)
be written
on the light-hand side is extended over the whole surface
It is easily seen
<7, and n l is the u^-component of the outward normal vector n.
that (3) holds also in the case whore tin* boundary surface cr is not convex, in
which case a straight line parallel to the j^-axis may intersect the surface in a
greater but even number of points. Similarly, if g(t l9 x 29 a* 3 ), h(x lt x 2 ,x 3 ) arc
two other given functions of the space coordinates we also have
wlieic the integration
1
/'"-/"*
(3')
If, in
field
particular, /,
(a t )
we
and h arc the components a^x),
by addition of the throe equations
g,
get
2(
a 3( x
a vector
(3), (3')
V
or
J
V
diva dV
^-
da
J (a n)
o
a n da,
(4)
where a n is the component of a in the direction of the outward normal. The equation (4) which enables us to transform the volume integral on the left-hand side
into a surface integral represents G aura's theorem (IV. 188).
359560
APPENDIXES
372
if
Further,
=/
*u
^2
tK (x) is
V>
**
a three-dimensional tensor
* from ( 3 > and < 3/
field
of rank
2,
we
get, putting
n K )rfa,
F
i
(5)
the equation (IV. 192).
e.
The transformation equations
2.
densities 8 l
and s^
for the four-current density
the transformation equations connecting the four -current
in two systems of mertia>$ and S' must be .such that the equation
According to (V.
4, 4')
(1)
gj-O
a consequence of the equation
is
<n
The transformation must
for all possible charge arid current distributions
fore bo of the form
there-
9
59 3 59}
SQ'-ffQ
4/
2'
Jt\ 5 l 5
i
where the functions /t must
for all variations of the
(2)
satisfy the relations
twenty variables
sl
to the restriction
<9
and
.9^
c)s k /cxi
which
arc*
subjected
a
"
/^v
\*7
j,t
Multiplying (4) by a Lagrangiari factor A, which may
(# t ), and subtracting this equation from (3), we get the equation
be a function of the variables
which must now hold
sk
By
j.
for arbitrary independent variations of the variables s l
variation of the variables s k> we get
^,,
or,
and
(6)
we
get
by
is
*?
m
,?
k in this equation
we
we must have
ax*
*;*<
get
A must be independent of the variables
of(6 '>
(6')
symmetrical in k and
dX
Taking
14),
differentiation with respect to
since the left-hand side
(6)
f )8 t ,.
= A(,K*
|(J
^A;
i.e.
A(
using the orthogonality relations (IV.
From
and
s % also.
Hence we get by integration
APPENDIXES
where the
sv
0, it
f$ lk
and A are constants. If the electric charge density is zero in S, i.e.
Hence J3tk =- and the transformations (2) take
must, also be zero in S'.
the form
tt
*t
From
373
(IV. 11), (V.
3),
and
(8)
we now
/o\
^ik 8 kget
= A ^s
- A (l-ttVc
2
fc ,
or
p'
(l-w
/2
/c
(9)
).
/>
the same relativity argument as was used in connexion with the equations
8, 14) it now follows that the constant A must have the value
By
(II.
A=l,
and
(8)
density
3.
(10)
then leads to the transformation (V
a four-vector.
5),
showing that the four-current
is
Plane waves in a homogeneous isotropic substance
In the rest system of a homogeneous isotropic substance with electric and
J -and JJL and p
Maxwell's equations (VII. 31, 32)
magnetic constants
be
written
may
_
cE
diyH
dlvE
ccurlH,
/xH
D-
0>
B^
eE,
(1)
cc-urlE,
(2)
(3)
/xH.
In the case of a plane wave with wave planes perpendicular to the .r-axis of a
Cartesian system of coordinates (x,y,z) the field vectors are functions of x and t
only. From 1 and from the ^-component of the vector equation (2) we therefore
(
get
w _
f=te =E*- H'-*
*3r
'
and
since constant fields are of
no importance
in optics
we may put
Et = H,~
The
y-
and ^-components of the equation
dEy
(4)
give
(2)
dH
8EZ
8H,
Differentiating one of the sets of equations (5) or (6) with respect to t and using
we see that each of the functions v z v z satisfies the wave
E E H H
the other set
equation
?
w=
where
is
a constant
The general
solution of the
^0
-~
VM
wave equation
(8)
(7) is
whero/j and/2 are arbitrary functions. The two terms in this expression represent
plane waves travelling with velocity w in the directions of the positive
APPENDIXES
374
.r-axis, respectively. Thus considering a
direction of the positive r-axis, we can put
and negative
--=
EZ
-*/>),
"*/(
~-
wave
travelling in the
c-*g(t-jt/w),
(9)
where/ and # are arbitrary functions of the argument t~xjw. From
get by integration
lfy
-fj.-*g(t-jr/w),
.-
(6)
wo then
(10)
p-*f(t-x/w).
The equations (4), (9), and (10) represent the most general expression for a plane
\va\e moving in the direction of the positive r-axis
Introducing three unit
tors
v<1
n
e
e< 3
these equations
may
E
(1,0,0),
(1)
(0,1,0),
>
(0,0,1),
be written
*-*/(/
.-.
-(x
(x n)/V)e (>)
n)ftt')ew-] c-*g(i
-n-lg(t- (x n)/M')e
(1
M-/A-V('-( x
n)e<
2
>,
fonn they remain tine if the system of (Coordinates i.s rotated
so that n does not lie in the direction of the new r-axis
e (1) and e^ are then
and
in this veetoi
aibitiar \ but constant unit vectors peipeiidicular to eaeli other ancl to the
t
wave
normal n
Transformation
4.
by
of the gravitational field variables y LK y x>
,
a change of coordinates inside a definite
The ti.msfoimation (VI II
x'^x'^t*),
=-
j'*
/(i'),
t/c
(1)
'
system of reference,
1-
-^
TTT
4
C^t'
\\e get
system of reference
,s^si(^nis inside the s.unc
/l
th
Thus
59)
connecting two dilferent coordinate
is ehnincte) i/ed by the equations
i
0.
M4
(2)
from (VIII. 57)
-
y[i
f
loin u Inch
*[&iyim
a[4<//4
4(if/Ai
J^ii).
(2')
we obtain
Further,
i-e.
Thus y LK transforms
(4)
ojajfy^
yi'ic
like
a spatial tensoi and
r/<7
--
y LK
djc'dx* is invariant
under
the transformation (1).
The transformation (VIII. 103)
x'<
---
ji
l
,
x'*
-/(.r
1
)
(5)
APPENDIXES
is
375
a special transformation (1) in which only the rate and setting of the coordinate
In this case we have
clocks are changed
<--g-S-i.
Using
this
m (VIII.
55) with
-=.-
we
4,
aj+cx 4 ai
If wo
we
get
c*X
0,
1.
want the gravitational vector potentials y[
get fiom
(3)
(7)
to be zeio in the piimrd system
the conditions
Mi
%}>*-
winch by means of
(7)
r^M
2 x/ c3 )
-J
2 x /(l2 )
lead to the equations (VIII
*V
,
___?!
()
'
109)
c^^V(H 2 X /c-)ca
= ^ =
-^~
/ii)
~'<J\\
Putting
<r t
we
get from
(2), (2
),
(6),
and
(11)
\J(
(S)
-<!
It
is
*1
now easily seen that the opeiator -
tion (5), tor
-fcr t
mv aiiant undei
the transforma-
we have
rV
winch togethci with
(12)
and
_i.
c'*'
(8)
CI K
c'r*-
il
c'i
'
gives
K ^r / +
.
is
straightforw.ird calculation then,
<I
(13 )
,^..
shows that the quantity (VIII. 110) which
can be ui itten
tiansforms according to the simple law
under the transformation (5), e each component of th(^ spatial tensor OJ IK is
by the intio b<tweon the rates of tho coordinate clocks in the two
i
multiplied
systems.
5.
Dual tensors in a three-dimensional space
In a three-dimensional space with a positive definite metric,
the quantities
1
*<*A
=-
fi
vy8 [KA
e.
--
\y LK
0,
...
(1)
are the covanant components of an antisymmetric pseudo-tensor if S
three-dimensional Levi-Civita symbol defined at the end of 43. Remembering
APPENDIXES
376
that the determinant |y *| formed by means of the contravanant components of
the metric tensor is equal to y" 1 we find that the contra variant components of the
pseudo -tensor (1) are
l
yU
ylK
ylA
The covariant components of the axml vector
--
IK
are
now
and the corresponding
components aro
contruvariaiit
H'-Je*l/ =
^8,
lrt
For W, and H' we thus get
tlio
an antisymmetric tensor
LJX
H dual to
>fA
ff A
l
(4)
following explicit expressions
(5)
n ,Hn ,Hn
(H*,H*,H*)~^(H
which show that the iclations reciprocal to
//
From two
eW
(3)
//,
A,
and
(4)
=- c
lKA
)t
are
H\
(5')
we can
build the antisymmetrical tensor c lK
and the corresponding dual axial vector is the vector product
vectors
l
a*, 6
tensor dual to an antisymmetrical tensor
F=
If
H ~ ^TT
l
^wA #KA
ls
(6)
_L
Vy
le
KA
VLK ^
^-l3
1KA
d ^y Hi
d&
means of
(5)
Similarly to
iK
for,
L^
of rank 3
is
a pseudo -scalar
^A.
tne vector dual to the tensor
dual to the curl of the tensor
corresponds the axial vector curia
dx K
dr*
The
-i
___
axb
of the two vectors whose components aro obtained by
the tensor curl, a
a Kb
a bK
(7)
IK
the divergence of
according to (X. 75),
is
we have
dH *
lffA
2Vy
fix 1
1
^}Similarly
6.
by means of
(5')
and
(2)
we
(8)
get
The condition for flat space
A flat space is defined as a space in which it is possible to introduce a system of
coordinates which
obviously have
is
geodesic at every point.
D
^iWm =
If this condition
is
fulfilled
we
(1)
APPENDIXES
377
at every point, where -#t jy m is the curvature tensor defined by (IX. 99). Conversely,
shall now show that ( I ) also represents a sufficient condition for flatness.
we
vector field a l (x)
is
called stationary if
the equation
it satisfies
at every point where T\r (x) is the ChnstofTel symbol. The equations (2) represent
a set of differential equations for the functions a i (x) which have solutions only if
the integrabihty conditions
f
dx m dx l
are satisfied.
The conditions
(3)
may
also be written
-** =
The integrabihty conditions
are thus fulfilled
be integrated and the solution
is
given.
is
da*
if (1)
holds,
uniquely determined
Since
if
and
o.
0')
in this case (2)
can
the vector o t (P) at a point
= ^dx = -TLa'dx
ox
l
1
,
a stationary vector field are, according to (IX. 62), obtained by
This is in accordance with (IX. 104), which shows in
displacements.
parallel
the case of a vanishing curvature tensor that the result of a parallel displacement
the vectors
of a vector is unique and independent of the path along which the displacement is
"
performed.
Let us now consider four different solutions a$
{af^, af2) afg) aJ4) } of the
differential equations (2) determined by four given vectors af (P) at the arbitrary
point P. The vectors a\k} (P) must be taken to be linearly independent for
instance orthogonal to each other. It can then be shown that we get a trans1
l
k which leads to a
formation x /l
jc' (x
system of coordinates (x' ) geodesic everywhere if we choose the transformation coefficients
,
fc)
K) =
In the first place
may be written
it is
(*)
&>(*)
seen that the integrabihty conditions (IX. 14) which also
QW1
i
*
are satisfied by the expressions (4). In fact, since the equations (2) hold for each
of the vector fields afk)9 we have
doil
rf
(JOLT,
dx r
& - a?
datir) .
-*=
-^
which on account of the symmetry of FJ5 in the lower indices is symmetrical in
k and I.
From the Christoffel formulae (IX. 53) and from (6) we now get
=
Thus the
in the
Christoffel symbols
system of coordinates
tensor are thus constants,
and therefore
also dffyjdx'* vanish at every point
In this system the components of the metric
we are dealing with a flat space.
(x'*).
i.e.
o.
APPENDIXES
378
The
7.
action principle and the Hamiltonian equations for a
particle in
an arbitrary gravitational
field
Consider a freely falling particle in an arbitrary external gravitational field
For simplicity, we shall make use of the possibility, mentioned in 113, of introducing a time-orthogonal system of coordinates (#*) = (X cfym which y = g u = 0.
The connexion between the proper time r of the particle and the coordinate time t
L
is
then given by (VIII. 99),
e.
= dt(\ + 2 X / C
w = ylK u u*
dr
where
~~
u*/c
)*,
K
y iK x x
l
1S the sca l ar gravitais the square of the velocity of the particle, and %
x( x ^
tional potential According to (VIII, 85) the time-track of the particle is determined by the variational principle
-o
and, since the integrand is a homogeneous function of the four variables dx' /dX of
first degree, (2) is equivalent to a variational problem with only three dependent
variables x and with t as the independent variable. f
Hence the motion of the particle is determined by a vanationul principle
l
analogous to Hamilton's principle in Newtonian mechanics In fact, multiplying
factor
which vanish for t
t
'tr^c*, we # ^ ^ 01 a ^ variations 8r
by a constant
and t
t2
ja
where the Lagrangian L
is
L(ji^jc^t)dt
J L(ji^jc^t
0,
(3)
given by
The corresponding Euler equations
Lagrangian equations of motion in Newtonian
mechanics for a system with the generalized coordinates x
x (t) From (4) we
are then analogous to the
et
with
dL
and p given by (X.
t
Further,
12)
(>L
r
t See, for instance,
Berlin 1924, p. 174
~dr l
(5)
take the form
--
at
13),
U K U A ~ m dx
~dx~L
Thus, the Lagrangian equations
with d c pjdt given by (X.
<-yKA
"
~dx'
which shows that the gravitational force
Courant and
7)
CJL
is
given
Hilbert, Methoden der Mathematischen Physik I t
APPENDIXES
379
17) Using the general formula (VIII. 98) for dr/dt instead of (VIII. 99) in
one could in the same way find the expression for the gravitational force in a
general system of coordinates, where y ^ and, in particular, the formula (X. 18).
From the Lagrangian form (5) of the equations of motion we can now pass
over to the canonical Hamiltoman form by the usual procedure. According to
(6), the canonical ly conjugate momenta of the coordinates x are the covanant
by (X.
(4),
components p of the momentum of the
(6), we get for the Hamiltoman function
--
pi^-L
----
=
which
is
Thus, by means of
particle
c 2 (l-f 2x/c 2 )(H-2 x /c 2
mc
+ 2 x /c
(l
~w
(4)
and
/c )-i
(8)
),
identical with the expression (X. 22) for the energy of a particle in a
0.
In a static gravitational
gravitational field with y
of the motion. Solving (6) with respect to w we get
t
field,
is
a constant
t,
which allows us
We
It
H as a function of the
to express
got
is
'*
now
canonical
canonical variables x and p
l
and
(10)
)*.
easily verified that the equations (6)
are equivalent to the
(7)
Hamiltoman equations
*-"
<-%
and the whole Hamilton Jaeobi thecny of integration can then
also be applied
in this case
r (t) is compared with a
In the variation principle (3), the actual motion
virtual motion
which the particle has the coordinates Jc
.**(<) -j-8.r ($) at the
time /, i e the symbol 8 refers to a variation when; the time is kept constant.
shall now consider a more general variation by which the time is varied also, so that
the particle in the* varied motion has the coordinates
#'(0 + A# at the time
l
We
where A
may be regarded as an arbitrary infinitesimal function of
Neglecting terms of order higher than the first, we obviously have
t-{ A/,
A.r l =- 8.ri+j/At
(12)
and, consequently,
A
v*t
__
_ __
_
~~
A^
dt
__
_____
~
__ _
"~
~dt l
dt
t.
dt
dt
d&i
d&t
d&t
_-_
+J -A,^.__^_,
1
Ar
01
Thus,
\ve get foi
Hence,
and
any function L(x
^(L
dt)
L
9
jc
l
,
t)
+x
Ldt
&t.
(13)
of the variables x
&Ldt + Ldkt
Sx
fi
$L
cft
dt-\
At)
l
9
Jt
-^-dt
and
(15)
(16)
APPENDIXES
380
According to
(3)
and
we
(16),
thus have
**
A
J
Ldt^Q
(17)
provided that we choose
A*
On
account of
(8) this
^ and
(18)
2.
also be written
may
Now we
for
p^d^-b
Hdt
0.
--=-
(19)
have
r
dt)
//A
rf/
|-
AH dt ~ H d&t-\- l8H-{-~j- At] dt
(20)
dt
where we have made use of
Since the
A -variation
is
(14) applied to the function H.
8 -variation,
we may now impose
more general than the
the further condition
s u-
OJnL
oi \
("*)
/
U,
which means that the energy of the particle at any time has the same value in
the actual and the virtual motion. This does not necessarily mean that the energy
must be constant along the path. From
(20) arid (21)
we then
get
"=0
and
(19) reduces to
p,
dr
(22)
'i
(22) together with (21) and (18) correspond to Maupertuis's principle of least
action in its most general form
is equal to a constant K> along the actual path,
For a static field, the energy
<?
and Maupertuis's
principle states that the 'action' J
p dx
L
m first approximation
*i
has the same value for all virtual motions with the same energy E. In the static
may even be eliminated entirely from (22). Since p = mw and
case, the time
u>
dcr/dt
p dx
p u
and
(22)
we have
may
dt -
be written
mil* da/u
mu dcr
=-
p da
(&)
0,
where
(x\)
and
(jc)
&T
this
(23)
coordinates of the particle at the times
and
2t
respectively.
Putting
in (10),
p =
and solving with respect
{(J5/C
p-m;
to p,
we
get
C 2 -2mi; x }i(l-f2 x /c2)-J.
(24)
Henco, the variational principle takes the form
Ui)
(l
+ 2 x /c
)-4{(^/c)
~m?c 2 -2^ x }*d<T -
0,
(25)
APPENDIXES
381
E as well as the end points (x\) and (x^) are to be kept constant by the
A-vanation. In this form, the variation principle is analogous to Hertz's 'principle
of the straightest path' in Newtonian mechanics. In the limit of weak fields and
small velocities the integrand in (25) reduces to
where
V{2m (e-w
which
where
x)},
E-Jk Q c 2
-=
the integrand occurring in Hertz's principle.
If ^ = 0, the integrand in (25) is a constant and we get
is
da
0,
(26)
which shows that the path is a geodesic in the space defined by the metric tensor y
As remarked by Hertz, f the more general case, where x(x = 0, can formally
be treated also as a problem of determining a straightest path, but in a space
where the line element is defined by
t/c
dS
- TIK dx
dx K
=p
da 2
>
(27)
Such a treatment
is, however, quite formal, since the geometrical structure of the
space as determined by natural measuring sticks is described by y tK and not by FtK
.
8.
The connexion between the determinants of the space-time
metric tensor and the spatial metric tensor
According to (VIII.
64, 63)
we have
(1)
044
The determinant y may be written
in the
form
0i4
024
we multiply the elements of one of the
unchanged
columns by a common factor and add them to the corresponding elements of
another column we have
Since a determinant
on account of
t
is
if
(1).
H. Hertz, Die Prinzipien der Mechamk, 2nd
ed., Leipzig (1910).
APPENDIXES
382
If
we apply
the
same procedure
to the second
and the
011
012
013
014
021
022
023
024
031
032
033
034
041
042
043
044
third
W
/
Since both # 44 and g are negative, y must thus be positive and
V("0)
9.
The
Vy V(l+2 x /c
(-044> Vy
column we get
we have
(3)
).
with respect to
derivatives of the function
ffi
and
lm
and some identities containing these derivatives
According to (XI 132) the function
where
Hence, since
</j!
</'*
lh
\\Y
i*
can be written
B =
s
is <
and g tk are kept constant
m we have
~-
*-'"--
g rfm Y L
lr
in partial derivation
^^^
(2)
Mith respect to
61,4
From (IX
68')
we
got
(6)
Further, by contraction of (IX. 68) with respect to the indices k and
(7)
Hence
winch
(8)
is
symrnctrji al in
Thus, by means
and
?//
(cf. tlie
remarks following (XI
140)).
of (4)- (8), w(^ get
8A
B we consider a variation
The corirspondmg variation of B is
In order to find the corresponding dei i\ ativc of
variables gff for constant g lk
1
^dLSiirsr
87*
If, in
k,
the last term,
I, r,
m we get
we perform a cyclic permutation
which by means of (IX
Hence
of the
&B
___
rfm S(g 'rfk
of the four summation indices
gT\k
),
68) reduces to
SB
(10)
APPENDIXES
which together with
(9)
and
(3)
383
leads to the desired formula
*~)O
^-
V(-flO{r{m-i(8? r^+s^rf^-K^Tfc-flf^r;.)^}.
In the same way we can find the derivative of i* with respect to y l>H
easier to use the equation (XI 139) according to which
(ii)
However,
it is
The last term may be obtained from (11) by differentiation with respect to x k
while the first term is given by (IX. 114, 111) Besides the components of the metric
tensor and Christoffel symbols these expressions contain derivatives of the
Christoffel symbols. Since fl and therefore also the left-hand side of (12) does not
contain derivatives of Christoffel symbols, all terms on the right-hand .side containing such derivatives must cancel arid we may therefore omit them from the
beginning in our calculation. By differentiation of (11) with respect to .*,* we are
then left with terms containing derivatives of ^J( g}, g, gi m only By means of
(IX 69', 68, 51) these terms and therefore also the right-hand side of (12) may
be expressed in terms of components of the metric tensorand of Christoffel symbols
After a straightforward calculation one finds
,
0,1,
V(- sOl-i'&rmr-rfrrj^- i(0j*
-j^r^Mi^H- i\ mk )}- ^ m
(is)
The expressions (11) and (13) for the derivatives of with respect to g and g lm
are easily seen to be in accordance with the equations (XI 141, 142).
It we now substitute (11) and (13)
the last term $]{ on the right-hand side of
l
(XI. 101) and use the equation
* an
following from (IX 68), we get for
expression containing a large number of
cancel
in
teims, which, howevei,
pairs. The final result is therefore that the
quantities i*J are identically zero
~~
d
1
.
82
AUTHOR INDEX
Abraham, M.,
88, 160, 193,
204, 205.
Adams,
348.
-B 26
Anderson, C D
Airy,
Neddermeyer, S
Hertz,
381.
Neumann, C 356
Neumann, G 89.
Hoek, M
Hubble, E
91.
Bambridge, K T 90
Beauregard, O C. de,
D,
Hilbert,
170.
Bethe, H., 90.
358, 362, 368.
L., 362, 368
C 1, 15
Huyghens,
Ilhngworth,
K K
Ives,
10,
H E
Born,
75, 157, 194, 195.
Bradley, J., 25.
de Broghe, L 58, 105
89
Bucherer, A.
,
Campbell, W. W., 355.
,
Courant,
Curie,
Johot, F 91.
Jordan, E. B 90.
,
Dirac, P.
A M
195,
204,
91, 98.
FitzGeralcl, G. F., 28.
10, 19,
20
Fokker, A. D 170, 195.
Foucault, L., 10
89.
Kaufmann,
Kennedy, R J 28
Kmoshita, S 10
Galileo,
220.
89.
89.
Hasenohrl, F., 211.
Heisenberg, W., 195.
170
Langevin,
Laue, M. v
Lemaitre,
369
Lense,
89
348.
G E
327, 364,
Lewis, G
Livingston,
309, 333.
90
6.
Meitner, L 91
Michelson, A. A., 15, 19,24,
,
26, 28.
Miller,
D. C
28.
93, 105, 136,
139, 149, 160, 195, 203
,
Morley, E. W.,
326,
v 356
327
,
10
Smith, N
Sommerfeld,
190,
19,
26
258,
jr
90
75,
129,
144
Stark,
G R
G G
Tamm, J
Thirrmg,
220
10
10,
62
15
327
204, 205, 206.
317, 320.
Thomas, L W., 56
Tolman, C., 67, 204, 214.
341, 357, 370.
355.
Trumpler,
Walton, G T. S 89.
5.
Weber,
Weyl,H., 157, 309, 333, 368.
Weyssenhoff, J. v., 250.
Moller, C., 170,
299, 369.
de, 317, 356.
Synge, J
21, 22, 24,
29,30,40,46, 82, 193,
195, 258
Mmkowski,
Sitter,
Stokes,
28,
Maxwell, J C
Stilwell,
67
S
Lodge, O., 28
Lorentz, H.
Siege!,
Southerns,
317
Levi-Civita,
Seehger,
Serim,
175, 204, 258
Lavanchy, C
Ledermann,
330.
85, 258
Sagnac, G 64
Scheye, A 209.
Schwarzschild,
Mie, G., 194.
Gerlach,
Guye, Ch.
Fresnel, A.-J., 16
Freundhch, E., 348.
Fried lander, B., 320
Fnedl&nder, T., 320.
93, 139, 194
P 364, 369
Robertson,
Rosonfeld, L., 21, 186.
224.
F 338
Kohlrausch, R
Einstein, A., 30, 31, 32, 41,
46, 49, 82, 139, 211, 219,
258, 310, 313, 321, 327,
338, 348, 355, 356, 358
Eotvos, R v., 220.
Fizeau,
170
Kant,
91.
Dallenbach,
206
S., 91.
Rasetti, F., 91
Reissner, H., 333
378.
P.
H
M H
Pryce,
348
Klein,
89.
Papapetrou,
Pomcare,
85, 88
Cockcroft, J.
Ocehiahni,
Champion, F. C
Chazy, J 352.
219.
4,
91.
Phihpp,
Planck, M., 181, 211.
28
62
91.
John, S
Newton, I,
Pauh, W., 204, 327.
G., 286.
Blackett, P
Bohr, N., 346.
91
311, 314, 378.
Bergmann, P.
Humason,
Becker, R 160, 204
Belmfante, F. J 186
10.
17.
Herglotz, G., 179.
Hermann,
Yukawa,
Zeeman,
184.
P., 63, 220, 221.
SUBJECT INDEX
Aberration of light, 25, 62.
Action principle for a particle in a gravita-
Electrodynamics in a gravitational
tional field, 378, 380.
Addition of velocities,
field,
302, in stationary matter, 195, in uniformly moving bodies, 196, in vacuo,
139.
3, 52.
Affine tensors, 273,
Electromagnetic
field
141,
tensor,
196,
302.
Angular momentum, 110, 138, 169, 189.
Electrons, classical model of, 193, theory
Bianchi identities, 286.
Black -body radiation, 216
of, 20.
Energy, conservation
of, 163,
337.
gravitational, 340.
Centrifugal force, 4, 218, 317.
Chris toffel formulae, 273
three-index symbols, 273
kinetic, 70.
of a particle in
tional field, 294.
Clock, coordinate, 226, 235
transformations
paradox, 49, 258
rate of moving, 48, 247
standard, 33
Closed system, centie of mass
170,
energy, 163, 337.
Coordinates, Cartesian, 92, 231.
curvilinear, 228, 233
equivalent systems of, 321
Gaussian, 228.
geodesic, 274
238, 296
Deflexion of light in a gravitational held,
353
de Sitter universe, 362.
Divergence of a tensor, 127, 283
of a vector, 127, 283.
Doppler effect in de Sitter universe, 367
non-relativistic, 8-10.
relativistic, 62.
258, 290,
theoiy, 105.
field,
295,
a gravitational
the
special
stress,
momentum
of,
electromagnetic, 159, 307.
for general fields, 185.
kinetic, 136.
Mmkowski's, 202
for perfect fluids, 182, 300.
total, 161, 163.
Equivalence
of energy
and mass,
78;
principle of, 220, 264.
Equivalent systems of coordinates, 321.
Kuler equation, 184, 229.
Fizeau's experiment, 19
Flat space, condition for, 376.
Force, electromagnetic, 155, 156, 203, 205,
306.
fictitious, 4, 218,
219
gravitational, 291
transformation
of, 70,
73
Four-acceleration, 102
current density, 140, 141, 197, 302
force, 105,
295
momentum,
104, 289
ray velocity, 103
vector, 99, 266
velocity, 102, 288.
vector, 103
Fresnel's dragging coefficient, 16, 63.
wave number
Galilean transformation, 2, 250.
Gauge transformation, 144, 248.
Gauss's theorem, 128, 371.
lines, 228, 272
system of coordinates, 274.
Geometry, non-Euclidean, 226.
Gradient of a scalar, 126, 279.
Geodesic
Einstein universe, 357.
Elastic matter, 173.
energy density
Format's principle, 23, 308.
of,
Cosmological models, 356
Covariance of the laws of nature, 97, 265
Co van ant differentiation, 280
Curl, 126, 127, 283, 284
Cuivature tensor, 284, contracted forms
of, 286
in
Abraham's,
elastic, 176.
of,
momentum and
Dual tensor, 114, 270.
Dynamics of a particle
tensor,
204
Conservation of electric charge, 140, 197,
302
pheudo-Cartosian, 233
quasi -Gali lean, 342
time -orthogonal systems
Conohs force, 4, 218, 317
of, 28.
Energy-momentum
definition of, 163.
of
a stationary. gravita-
density,
175-81.
and
Gravitational
field,
stationary, 250, 294.
static,
250,
323;
SUBJECT INDEX
386
Gravitational Held equations, 310, linear
approximation
a paitiele
external gravitational field, 379
Hook's experiment, 1 7
Huygheris principle, 11.
Hyperbolic motion, 75
for
in
un
Poynting's vector,
Pseudo-tensor, 112, 270
Rate of moving clock
field,
Hay
monatomic gases, 215
Incoherent matter, 130, enorgy-momentum tensor for, 1 36, 300
Ideal
Inert lal system,
Interval, 99.
I.
Kronocker symbol, 94
velocity, transfoirnation of,
04
solution,
325,
anterior solution, 328
118; without
Simultaneity of events, 31, 33
Static giavitational holds with spherical
rota-
symmetry, 323
-
non-closed systems, 191
Systems with sphencal symmetry, 322
of refeience, general accelerated, 233,
Mans of a cloned system, 77
234
of a material paitic lo in a giavitational
290, in a system of meitia, 69
Meson fields, 184
Metric tensor, 228,
inertial, 1, 17
field,
experimental determination
of, 231,
237.
rigid,
253
unifoirnly lotating, 222, 240
Tensor, 108, 111, 269
ami psoudo -tensor fields, 125, 279
Thermodynamics, foui -dimensional
properties of the space-time, 235
space-time, 233
mulation
238
of,
Thomas
inertia, 69.
Tune track office
of, 71
Non-closed system, definition
in
uniformly
moving matter, 212
1 1 1, 138, 190
of a material particle in a
gravitational field, 290, in a system of
of force,
Momentum
for-
214
in stationary matter, 21
Michelson's experiment, 26
precession, 56, 121, 125
Time-oithogonal system of coordinates,
238, 296
particles
and
light rays,
244
of,
188
Parallel displacement of \octois, 276
Particle velocity, tiansforination of,
51, 52, 53
Perfect fluids, 181
Perihelion, advance
and
principle of mechanics, 1-4.
special piituiple of, 4
Schwarzschild's oxtonor
transformations, general, 41, 92.
infinitesimal, 117
transformation
15,
of inertia, 48, 49, 97
Lorentz contraction, 28, 44, 96
Moment
11,
Retardation of moving clocks in a system
Levi-Ci vita symbol, 113
Local systems of inertia,
spatial,
in a giavitational
247.
58
Reference points, 234
systems, ^eo systems- of reference.
forces
Relativity of centrifugal
Conolis forces, 317.
general pimciple of, 218
--
special, 36, 95.
successive, 53,
tion, 42, 118.
gravitational
dynamical, 296-8
Li&nard-Wiechert's, 149
retarded, 148, 315
Gravitational mass, density of, 344
shift of spectral lines, 346
Hamiltoman equations
of
elimination
Potentials,
313
of,
Vanational pimciple of electrodynamics,
157, for geodesies, 299, for gravitational fields, 333 , for time tracks of free
3,
paiticles
and
light rays, 244
Velocity of light in gravitational fields,
240, 308, in refractive media, 15, in
of,
348
Permanent gravitational fields, 221
Phase velocity, ttansfoii nation of, 8, 23
7'arwo, 10
Velocity of propagation of the energy, 164
ma lightwave,
Potentials, electromagnetic, 143
dynamical gravitational, 246
Work, 70
161,
206
You might also like
- General Relativity 2012 Harvey Reall Lecture Notes100% (1)General Relativity 2012 Harvey Reall Lecture Notes172 pages
- Advanced Topics in Quantum Field TheoryNo ratings yetAdvanced Topics in Quantum Field Theory131 pages
- Concepts in Theoretical Physics by Baumann PDF100% (1)Concepts in Theoretical Physics by Baumann PDF105 pages
- Gauge Theories Lecture Notes by Matthew de Angelis100% (2)Gauge Theories Lecture Notes by Matthew de Angelis84 pages
- The Theory of Complex Angular Momenta - Gribov Lectures On Theoretical Physics (CUP)100% (2)The Theory of Complex Angular Momenta - Gribov Lectures On Theoretical Physics (CUP)311 pages
- A First Course in General Relativity SolNo ratings yetA First Course in General Relativity Sol41 pages
- Bjorken, Drell - Relativistic Quantum MechanicsNo ratings yetBjorken, Drell - Relativistic Quantum Mechanics156 pages
- QFT Solutions (Including Peskin & Schroeder 3.1 and 3.4)100% (1)QFT Solutions (Including Peskin & Schroeder 3.1 and 3.4)19 pages
- An Introduction To Classical Field TheoryNo ratings yetAn Introduction To Classical Field Theory8 pages
- ANN PHYS-Wheeler-Physics Classical Is GeometryNo ratings yetANN PHYS-Wheeler-Physics Classical Is Geometry79 pages
- CECOTTI Statistical Mechanics A Concise Advanced TextbookNo ratings yetCECOTTI Statistical Mechanics A Concise Advanced Textbook365 pages
- Greiner W. Thermodynamics and Statistical MechanicsNo ratings yetGreiner W. Thermodynamics and Statistical Mechanics463 pages
- Spacetime Compactification in General RelativityNo ratings yetSpacetime Compactification in General Relativity36 pages
- Matrix Product States For Gauge Field Theories: PACS NumbersNo ratings yetMatrix Product States For Gauge Field Theories: PACS Numbers10 pages
- Fundamentals of Mathematical Physics Edgar A KrautNo ratings yetFundamentals of Mathematical Physics Edgar A Kraut21 pages
- Introduction To Quantum Field Theory: Marina Von Steinkirch State University of New York at Stony Brook March 3, 2011100% (2)Introduction To Quantum Field Theory: Marina Von Steinkirch State University of New York at Stony Brook March 3, 2011121 pages
- Gregory L. Baker, Jerry P. Gollub Chaotic Dynamics - An Introduction PDF100% (1)Gregory L. Baker, Jerry P. Gollub Chaotic Dynamics - An Introduction PDF134 pages
- (Interdisciplinary Applied Mathematics 1) Martin C. Gutzwiller (Auth.) - Chaos in Classical and Quantum Mechanics-Springer-Verlag New York (1990)No ratings yet(Interdisciplinary Applied Mathematics 1) Martin C. Gutzwiller (Auth.) - Chaos in Classical and Quantum Mechanics-Springer-Verlag New York (1990)445 pages
- A Derivation of The Equipartition Theorem: More Chapter 8, #2No ratings yetA Derivation of The Equipartition Theorem: More Chapter 8, #22 pages
- Arxiv Sunil Mukhi String Theory Review 1110.2569No ratings yetArxiv Sunil Mukhi String Theory Review 1110.256945 pages
- Quantum Theory As An Emergent Phenomenon The Statistical Mechanics of Matrix Models As The Precursor of Quantum Field Theory (Stephen L. Adler) (Z-Library)No ratings yetQuantum Theory As An Emergent Phenomenon The Statistical Mechanics of Matrix Models As The Precursor of Quantum Field Theory (Stephen L. Adler) (Z-Library)238 pages
- Histories of The Electron - The Birth of MicrophysicsNo ratings yetHistories of The Electron - The Birth of Microphysics529 pages
- Jean Zinn-Justin-Quantum Field Theory and Critical Phenomena (Fourth Edition, 2002) - CLARENDON PRESS, OXFORD 2002 (2002)100% (1)Jean Zinn-Justin-Quantum Field Theory and Critical Phenomena (Fourth Edition, 2002) - CLARENDON PRESS, OXFORD 2002 (2002)1,219 pages
- Quantum Mechanics, An Introduction - Walter Greiner100% (10)Quantum Mechanics, An Introduction - Walter Greiner512 pages
- Advanced Topics in Quantum Field Theory A Lecture Course by Shifman M.No ratings yetAdvanced Topics in Quantum Field Theory A Lecture Course by Shifman M.642 pages
- Libro The Theory of Relativity Moller 1952 p.366No ratings yetLibro The Theory of Relativity Moller 1952 p.366409 pages
- DH XVR7104 08 4KL B X - Datasheet - 20201027No ratings yetDH XVR7104 08 4KL B X - Datasheet - 202010273 pages
- Design of Concrete Armour Layers: J.W. Van Der MeerNo ratings yetDesign of Concrete Armour Layers: J.W. Van Der Meer9 pages
- Od Dog, To, 2024: Commerce Examination, July, 2024No ratings yetOd Dog, To, 2024: Commerce Examination, July, 20244 pages
- IUPAC Nomenclature - DPP 03 (Of Lec 05) - Prayas JEE 2025No ratings yetIUPAC Nomenclature - DPP 03 (Of Lec 05) - Prayas JEE 20253 pages
- Trigonometry - I: Inverse Trigonometric FunctionsNo ratings yetTrigonometry - I: Inverse Trigonometric Functions37 pages
- Demystify OpenAI Triton Fkong' Tech BlogNo ratings yetDemystify OpenAI Triton Fkong' Tech Blog17 pages
- School Report On Performance of Students in Quarterly AssessmentNo ratings yetSchool Report On Performance of Students in Quarterly Assessment2 pages
- Cardiac Invalidism and Perceived Family Support Among Patients With Myocardial InfarctionNo ratings yetCardiac Invalidism and Perceived Family Support Among Patients With Myocardial Infarction3 pages
- Amurru Akkadian - A Linguistic Study - Shlomo Izre'el - 1991No ratings yetAmurru Akkadian - A Linguistic Study - Shlomo Izre'el - 1991257 pages
- Handbook of Advanced Chromatography Mass Spectrometry Techniques 1st Edition Michal Holcapek Instant Download100% (2)Handbook of Advanced Chromatography Mass Spectrometry Techniques 1st Edition Michal Holcapek Instant Download59 pages